or $F_{i}$ is a real copy and $F_{i}\in\mathscr{S}_{G_{i}}$.

Evidently $\mathscr{D}$ has the properties ($B2$) – ($B4$) of an acceptable
big cycle in $(H,\equiv^{H},\mathscr{H}_{\bullet}^{+})$. Moreover, if some
copy $G_{\star}\in\mathscr{H}_{\bullet}^{+}$ appears twice in consecutive
positions of $\mathscr{D}$, then $G_{\star}$ is a real copy and we get a
contradiction to the fact that $\mathscr{H}_{G_{\star}}$ is scattered in
$G_{\star}$. So altogether, $\mathscr{D}$ is a big cycle.

The acceptability of $\mathscr{D}$ and
$\mathrm{ord}(\mathscr{D})=\mathrm{ord}(\mathscr{C})\leq g$ are clear. Thus
our assumption $\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}_{\bullet}^{+})>g$
leads to a supreme copy $G_{\star}$ of $\mathscr{D}$. Recall that $G_{\star}$
is a real copy by Remark 8.8(0 ). Set
$K=\\{i\in{\mathds{Z}}/n{\mathds{Z}}\text{ and }G_{i}\neq G_{\star}\\}$ and
let the family of $G_{\star}$-pieces $\\{P_{i}\colon i\in K\\}$ exemplify the
supremacy of $G_{\star}$ in $\mathscr{D}$. Starting with $\mathscr{C}$ we can
still replace every copy $F_{i}$ with $i\in K$ by the piece $P_{i}$. If $n=2$
this shows immediately that $\mathscr{C}$ has a supreme copy and for $n\geq 3$
Lemma 8.22( ) ‣ 8.22 tells us that we obtain an acceptable big cycle
$\mathscr{E}$ in this manner. Due to Claim 9.13 $\mathscr{E}$ has a supreme
copy and by Lemma 8.22( ) ‣ 8.22 so does $\mathscr{C}$. ∎

###### Proof of Lemma 9.1.

We assume that the reader has the eight-step description (0 ) – (0 ) of
$\mathrm{Ext}(\Phi,\Psi)$ given immediately before Lemma 6.11 on the desk and
without further explanation we use the same notation as there. Let us remark
that the pretrain $(G,\equiv^{G})$ generated in Step (0 ) satisfies
$\mathfrak{girth}\bigl{(}G,\equiv^{G}\bigr{)}>g$, whence
$\mathrm{girth}(M)>g$. So the application of $\Psi_{r^{e(X)}}$ in Step (0 ) is
justified, meaning that the system $(H,\equiv^{H},\mathscr{H}^{+})$ does
indeed exist. It remains to establish that its $\mathfrak{Girth}$ exceeds $g$.

Let $\equiv^{N}$ be the equivalence relation on $E(N)$ whose equivalence
classes are single edges. By Step (0 ) the hypergraph $H$ is living in $N$ and
the pretrain $(H,\equiv^{H})$ is derived from $(N,\equiv^{N})$. Similarly, the
first part of Step (0 ) says that the system $\mathscr{H}_{\bullet}$ is
derived from $\mathscr{N}$. Due to $\mathrm{Girth}(N,\mathscr{N}^{+})>g$ and
Lemma 8.18 we know $\mathfrak{Girth}(N,\equiv^{N},\mathscr{N}^{+})>g$ and,
therefore, Lemma 9.6 discloses
$\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}_{\bullet}^{+})>g$.

For every copy $(G_{\star},\equiv^{G_{\star}})\in\mathscr{H}_{\bullet}$ the
$|\mathscr{X}|$ standard copies of $(F,\equiv^{F})$ form a scattered system
(cf. Lemma 6.8) and thus $\mathscr{H}$ is scattered in
$\mathscr{H}_{\bullet}$. Now
$\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>g$ follows from Lemma 9.12. ∎

###  9.2. $\mathfrak{Girth}$ preservation

The next result and its proof share strong similarities with the material in
Section 5. However, the focus there was on making incremental progress,
whereas here we only care about maintaining what we already have.

######  Proposition 9.14.

Suppose that $g\geq 2$ and that $\Xi$ is a partite lemma

  1. $\bullet$

applicable to $k$-partite $k$-uniform pretrains $(B,\equiv^{B})$ with
$\mathfrak{girth}(B,\equiv^{B})>g$

  2. $\bullet$

and producing $k$-partite $k$-uniform systems of pretrains
$(H,\equiv^{H},\mathscr{H})$ with

$\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>g\,.$

If $\Phi$ denotes an arbitrary linear Ramsey construction for hypergraphs
generating systems of strongly induced copies, then the pretrain construction
$\mathrm{PC}(\Phi,\Xi)$

  1. $\bullet$

applies to all pretrains $(F,\equiv^{F})$ with
$\mathfrak{girth}(F,\equiv^{F})>g$

  2. $\bullet$

and delivers systems of pretrains $(H,\equiv^{H},\mathscr{H})$ with
$\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>g$.

One may briefly wonder why this is a useful result. After all, the main
property $\mathrm{PC}(\Phi,\Xi)$ is shown to have is already assumed for
$\Xi$. Indicating only one example of a successful application of Proposition
9.14 we would like to point out that by Lemma 3.4 the copies in the systems
produced by $\mathrm{PC}(\Phi,\Xi)$ will always have clean intersections,
while this need not be the case for the systems generated by $\Xi$ (whose
copies can intersect in entire wagons). For this reason,
$\mathrm{PC}(\Phi,\Xi)$ can have more desirable properties than $\Xi$ itself.
As usual, the main work going into the proof of Proposition 9.14 concerns a
picturesque lemma.

######  Lemma 9.15.

Suppose that $g\geq 2$ and that

$(\Sigma,\equiv^{\Sigma},\mathscr{Q},\psi_{\Sigma})=(\Pi,\equiv^{\Pi},\mathscr{P},\psi_{\Pi})\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr+\cr\times\crcr}}}}(H,\equiv^{H},\mathscr{H})$
(9.2)

holds for two pretrain pictures $(\Pi,\equiv^{\Pi},\mathscr{P},\psi_{\Pi})$
and $(\Sigma,\equiv^{\Sigma},\mathscr{Q},\psi_{\Sigma})$ over a linear system
$(G,\mathscr{G})$ with strongly induced copies and for a $k$-partite
$k$-uniform system of pretrains $(H,\equiv^{H},\mathscr{H})$. If

$\mathfrak{Girth}(\Pi,\equiv^{\Pi},\mathscr{P}^{+})>g\quad\text{ and
}\quad\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>g\,,$

then $\mathfrak{Girth}(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})>g$.

###### Proof.

The picturesque statement in the proof of Lemma 3.11 reveals that $\Sigma$ is
a linear hypergraph. So it remains to establish

  1. ( )

that any two wagons of $(\Sigma,\equiv^{\Sigma})$ intersect in at most one
vertex

  2. ( )

and that every acceptable big cycle $\mathscr{C}$ in
$(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$ with $\mathrm{ord}(\mathscr{C})\leq
g$ has a supreme copy.

Stage A: First observations. The assumption that the copies in $\mathscr{G}$
be strongly induced will be used in the following way.

######  Claim 9.16.

If $F_{\star}\in\mathscr{Q}^{+}$ and $x,y\in V(F_{\star})\cap V(H)$ are two
distinct vertices, then there exists an edge $f$ with $x,y\in f\in
E(F_{\star})\cap E(H)$.

###### Proof.

Let the amalgamation (9.2) be constructed over the edge $e\in E(G)$. Since
$F_{\star}$ intersects every music line of the pretrain picture
$(\Sigma,\equiv^{\Sigma},\mathscr{Q},\psi_{\Sigma})$ at most once, the
vertices $\psi_{\Sigma}(x),\psi_{\Sigma}(y)\in
e\cap\psi_{\Sigma}[V(F_{\star})]$ are distinct. So if
$F_{\star}=f_{\star}^{+}$ is an edge copy, then the linearity of $G$ yields
$\psi_{\Sigma}[f_{\star}]=e$, for which reason $f=f_{\star}$ is as desired. If
the copy $F_{\star}$ is real, then $\psi_{\Sigma}$ projects it into
$\mathscr{G}$ and thus onto a strongly induced subhypergraph of $G$. In
particular, this projection needs to contain the edge $e$. The edge $f\in
E(F_{\star})$ projected to $e$ has the desired property. ∎

We proceed by transferring the hypothesis
$\mathfrak{Girth}(\Pi,\equiv^{\Pi},\mathscr{P}^{+})>g$ to standard copies.
More precisely, we deal with the special case of cycles in
$(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$ all of whose copies belong to the
same standard copy of the previous picture.

######  Claim 9.17.

Let $(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}_{\star})$ be a standard
copy. If all copies of an acceptable big cycle $\mathscr{C}$ in
$(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$ with $\mathrm{ord}(\mathscr{C})\leq
g$ belong to $\mathscr{P}_{\star}^{+}$, then $\mathscr{C}$ possesses a supreme
copy.

###### Proof.

Let $\mathscr{C}=F_{1}q_{1}\ldots F_{n}q_{n}$. Strictly speaking,
$\mathscr{C}$ does not need to be a big cycle in the pretrain system
$(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}_{\star}^{+})$, for the wagon
connectors of $\mathscr{C}$ are wagons of $\equiv^{\Sigma}$ rather than wagons
of $\equiv^{\Pi_{\star}}$. However, ($B4$) shows that every wagon among
$q_{1},\ldots,q_{n}$ contracts to $(\Pi_{\star},\equiv^{\Pi_{\star}})$. Thus
we can define

$\overline{q}_{i}=\begin{cases}\text{the contraction of $q_{i}$ to
$(\Pi_{\star},\equiv^{\Pi_{\star}})$,}&\text{ if $q_{i}$ is a wagon}\cr
q_{i},&\text{ if $q_{i}$ is a vertex}\end{cases}$

for every $i\in{\mathds{Z}}/n{\mathds{Z}}$ and then

$\overline{\mathscr{C}}=F_{1}\overline{q}_{1}\ldots F_{n}\overline{q}_{n}$

will be a big cycle in the system of pretrains
$(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}_{\star}^{+})$ with
$\mathrm{ord}(\overline{\mathscr{C}})=\mathrm{ord}(\mathscr{C})\leq g$. It is
easily confirmed that $\overline{\mathscr{C}}$ is acceptable and, therefore,
$\overline{\mathscr{C}}$ possesses a supreme copy
$F_{\star}\in\\{F_{1},\dots,F_{n}\\}$. Let

$\overline{\mathscr{M}}=\\{\overline{P}_{i}\colon
i\in{\mathds{Z}}/n{\mathds{Z}}\text{ and }F_{i}\neq F_{\star}\\}$

be a family of $F_{\star}$-pieces with respect to the ambient system
$(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}_{\star}^{+})$ that exemplifies
the supremacy of $F_{\star}$ in $\overline{\mathscr{C}}$.

We intend to show that $F_{\star}$ is a supreme copy of $\mathscr{C}$ as well
and to this end we need to convert $\overline{\mathscr{M}}$ into an
appropriate family of $F_{\star}$-pieces with respect to
$(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$. For every index $i$ such that
$\overline{P}_{i}$ is short we just set $P_{i}=\overline{P}_{i}$. If
$\overline{P}_{i}=(f^{\prime}_{i})^{+}\overline{W}_{i}(f^{\prime\prime}_{i})^{+}$
is long, however, then we put
$P_{i}=(f^{\prime}_{i})^{+}W_{i}(f^{\prime\prime}_{i})^{+}$, where $W_{i}$
denotes the wagon of $(\Sigma,\equiv^{\Sigma})$ contracting to
$\overline{W}_{i}$.

It will turn out that

$\mathscr{M}=\\{P_{i}\colon i\in{\mathds{Z}}/n{\mathds{Z}}\text{ and
}F_{i}\neq F_{\star}\\}$

is the desired family of $F_{\star}$-pieces. The main point to be checked here
is that the long pieces in this family still satisfy Definition 8.5( ) ‣ 8.5(
)( ) ‣ ( ) ‣ 8.5 with respect to the larger pretrain
$(\Sigma,\equiv^{\Sigma})$. Assume for the sake of contradiction that $P_{i}$
is a long piece and that some edge $f$ in $E(\Sigma)\smallsetminus
E(\Pi_{\star})$ contains the connectors $q_{i-1}$, $q_{i}$. Due to the
linearity of $G$ this requires $f\in E(H)$. But now Lemma 8.14 applied to the
system $(H,\equiv^{H},\mathscr{H}^{+})$ shows that $f$ belongs to the copy
$\Pi^{e}_{\star}\in\mathscr{H}$ extended by $\Pi_{\star}$, contrary to
$f\not\in E(\Pi_{\star})$. This concludes the proof that $F_{\star}$ is a
supreme copy of $\mathscr{C}$. ∎

Next we study a peculiar kind of big cycles in our amalgamation that will
occasionally require a special treatment later.

######  Claim 9.18.

Let $(\Pi_{\star},\mathscr{P}_{\star})$ and
$(\Pi_{\star\star},\mathscr{P}_{\star\star})$ be two distinct standard copies
of $(\Pi,\mathscr{P})$. If

$\mathscr{C}=F_{1}W_{\star}F_{2}xF_{3}W_{\star\star}F_{4}y$

is a big cycle in $(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$ such that the
connectors $x$, $y$ are vertices, $W_{\star}$, $W_{\star\star}$ are wagons,
and $F_{1},F_{2}\in\mathscr{P}_{\star}^{+}$ as well as
$F_{3},F_{4}\in\mathscr{P}_{\star\star}^{+}$, then there exists an edge $f\in
E(H)$ with $x,y\in f$.

$y$$x$$F_{1}$$F_{2}$$W_{\star}$$(\Pi_{\star},\mathscr{P}_{\star})$$F_{4}$$F_{3}$$W_{\star\star}$$(\Pi_{\star\star},\mathscr{P}_{\star\star})$$(H,\equiv^{H},\mathscr{H})$
Figure 9.3. The configuration analysed in Claim 9.18

###### Proof.

Notice that $x,y\in V(H)$. Due to the linearity of the pretrain
$(H,\equiv^{H})$ it cannot be the case that $W_{\star}$, $W_{\star\star}$
contract to wagons $W^{H}_{\star}$, $W^{H}_{\star\star}$ of $(H,\equiv^{H})$
with $x,y\in V(W^{H}_{\star})\cap V(W^{H}_{\star\star})$. By symmetry we may
therefore suppose that

if the contraction $W^{H}_{\star}$ exists, then $\\{x,y\\}\not\subseteq
V(W^{H}_{\star})$. (9.3)

Let $\Pi^{e}_{\star},\Pi^{e}_{\star\star}\in\mathscr{H}$ be the copies
extended by $\Pi_{\star}$, $\Pi_{\star\star}$. Now
$\mathscr{D}=\Pi^{e}_{\star}x\Pi^{e}_{\star\star}y$ is a big cycle in
$(H,\mathscr{H}^{+})$. If either of its copies is an edge copy, then the
existence of the desired edge $f$ is clear, so we can assume that
$\Pi^{e}_{\star}$, $\Pi^{e}_{\star\star}$ are both real.

Owing to $\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>g\geq 2$ Lemma 8.13
tells us that $\Pi^{e}_{\star}$ is a supreme copy of $\mathscr{D}$. Let the
$\Pi^{e}_{\star}$-piece $P$ exemplify this fact. If $P=f^{+}$ is short we have
found the desired edge $f$. It remains to consider the case that
$P=(f^{\prime})^{+}W^{H}(f^{\prime\prime})^{+}$ is long, where, let us
emphasise, $W^{H}$ is a wagon of $(H,\equiv^{H})$.

By $x,y\in V(W^{H})$ and (9.3) it cannot be the case that
$W^{H}=W^{H}_{\star}$. Thus the wagon $W$ of $(\Sigma,\equiv^{\Sigma})$
contracting to $W^{H}$ is distinct from $W_{\star}$. A fortiori the
contractions $\overline{W}$, $\overline{W}_{\star}$ of $W$, $W_{\star}$ to
$(\Pi_{\star},\equiv^{\Pi_{\star}})$ are distinct as well and, therefore,

$\mathscr{E}=F_{1}\overline{W}_{\star}F_{2}x(f^{\prime})^{+}\overline{W}(f^{\prime\prime})^{+}y$

(see Figure 9.4) is a big cycle in
$(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}_{\star}^{+})$.

$y$$x$$F_{1}$$F_{2}$$\overline{W}_{\star}$$(\Pi_{\star},\mathscr{P}_{\star})$$f^{\prime\prime}$$f^{\prime}$$\overline{W}$
Figure 9.4. The big cycle $\mathscr{E}$.

Clearly, it satisfies the acceptability condition ($A1$), and ($A2$) could
only fail if the desired edge $f$ exists. Moreover, ($A3$) holds due to the
fact that the only wagon of the linear pretrain
$(\Pi_{\star},\equiv^{\Pi_{\star}})$ that contains $x$ and $y$ is
$\overline{W}$ and, hence, a connector of $\mathscr{E}$.

Altogether we may assume that $\mathscr{E}$ is acceptable. Due to

$\mathfrak{Girth}(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}^{+})>g\geq
2=\mathrm{ord}(\mathscr{E})$

this implies that $\mathscr{E}$ has a supreme copy. By Fact 8.15 such a
supreme copy needs to contain both $x$ and $y$ and thus Claim 9.16 leads again
to the desired edge $f$. ∎

Stage B: Segmentations. When considering any (not necessarily acceptable) big
cycle $\mathscr{C}=F_{1}q_{1}\ldots F_{n}q_{n}$ in $(\Sigma,\equiv^{\Sigma})$
we can define segments of $\mathscr{C}$, their leaders, and segmentations of
$\mathscr{C}$ as in the proof of Lemma 5.2. Unless the copies of $\mathscr{C}$
are contained in a single standard copy of $(\Pi,\mathscr{P}^{+})$ we obtain a
segmentation

$\mathscr{C}=I_{1}r_{1}\ldots I_{t}r_{t}$

with $t\geq 2$ such that any two consecutive ones among the leaders
$\Pi^{e}_{1},\ldots,\Pi^{e}_{t}$ are distinct. As usual we want to pass to a
big cycle in $(H,\equiv^{H},\mathscr{H}^{+})$ by replacing every segment
$I_{\tau}$ by its leader $\Pi^{e}_{\tau}$.

The resulting cyclic sequence clearly has the properties ($B1$) – ($B3$).
However, the wagons among $r_{1},\ldots,r_{t}$ usually fail to be wagons of
$(H,\equiv^{H})$ and thus they are, pedantically speaking, not allowed to
serve as wagons of such cycles. Due to the clauses ( ) ‣ 7.3 and ( ) ‣ 7.3 of
Lemma 7.3, however, we know that such wagon connectors $r_{\tau}$ contract to
$(H,\equiv^{H})$. More precisely, if the leaders $\Pi^{e}_{\tau}$ and
$\Pi^{e}_{\tau+1}$ are in $\mathscr{H}$, then ( ) ‣ 7.3( )( ) ‣ ( ) ‣ 7.3
shows that they contain edges of $r_{\tau}$. Similarly, if only one of them is
in $\mathscr{H}$ while the other one is an edge copy, we appeal to ( ) ‣ 7.3(
)( ) ‣ ( ) ‣ 7.3 and the case that both are edge copies is unproblematic due
to ( ) ‣ 7.3.

Therefore we can define

$\overline{r}_{\tau}=\begin{cases}\text{the contraction of $r_{\tau}$ to
$(H,\equiv^{H})$}&\text{ if $r_{\tau}$ is a wagon}\cr r_{\tau}&\text{ if
$r_{\tau}$ is a vertex}\end{cases}$

for every $\tau\in{\mathds{Z}}/t{\mathds{Z}}$ and the cyclic sequence

$\mathscr{D}=\Pi^{e}_{1}\overline{r}_{1}\ldots\Pi^{e}_{t}\overline{r}_{t}$

will be a big cycle in $(H,\equiv^{H},\mathscr{H}^{+})$. Let us observe that
Fact 4.19 also holds for big cycles instead of cycles of copies, because its
proof is based on a computation that can be carried out in both scenarios.
Utilising the thus modified form of Fact 4.19 repeatedly we obtain
$\mathrm{ord}(\mathscr{D})\leq\mathrm{ord}(\mathscr{C})$. It will be
convenient to call $\mathscr{D}$ a reduct of $\mathscr{C}$.

Stage C: The proof of ( ) ‣ 9.2. Recall that by Fact 6.10 the wagons of linear
pretrains are strongly induced. Thus the following claim can be regarded as a
first step towards proving that $(\Sigma,\equiv^{\Sigma})$ is linear.

######  Claim 9.19.

Every wagon of $(\Sigma,\equiv^{\Sigma})$ is strongly induced.

###### Proof.

Let $W$ denote an arbitrary wagon of $(\Sigma,\equiv^{\Sigma})$. Due to the
definition of wagons we have $E(W)\neq\varnothing$ and no vertex of $W$ is
isolated. Hence it remains to check that if an edge $f\in E(\Sigma)$
intersects $V(W)$ in two distinct vertices $x$ and $y$, then $f\in V(W)$. To
this end we consider pairs of edges $e_{x},e_{y}\in E(W)$ with $x\in e_{x}$
and $y\in e_{y}$. If $e_{x}$ and $e_{y}$ can be chosen in such a way that any
two of the three edges $e_{x}$, $e_{y}$, and $f$ coincide, we conclude $f\in
E(W)$ either directly or by appealing to the linearity of the hypergraph
$\Sigma$. Assuming that this is not the case we obtain, for each choice of
$(e_{x},e_{y})$, a big cycle

$\mathscr{C}=f^{+}xe_{x}^{+}We_{y}^{+}y$

of order $2$. It should perhaps be pointed out that $\mathscr{C}$ is always
inacceptable, for the edge $f$ exemplifies that the wagon $W$ violates the
moreover-part of ($A2$). Nevertheless, $\mathscr{C}$ can have acceptable
reducts and we can learn something by looking at supreme copies of such
reducts.

If the edges $e_{x}$, $e_{y}$ can be selected in such a way that together with
$f$ they belong to a common standard copy
$(\Pi_{\star},\equiv^{\Pi_{\star}})$, then $f\in E(W)$ follows from the fact
that the contraction of $W$ to $(\Pi_{\star},\equiv^{\Pi_{\star}})$ is
strongly induced in $\Pi_{\star}$.

So we can suppose that such a choice of $e_{x}$, $e_{y}$ is impossible. Now
let these edges and a segmentation

$\mathscr{C}=I_{1}r_{1}\ldots I_{t}r_{t}$ (9.4)

be determined in such a manner that $t\in\\{2,3\\}$ is minimal. Write
$\mathscr{D}=\Pi^{e}_{1}\overline{r}_{1}\ldots\Pi^{e}_{t}\overline{r}_{t}$ for
the corresponding reduct. Up to symmetry one of the following three cases
occurs.

First Case. $\mathscr{D}=f^{+}x\Pi_{2}^{e}y$

Let $\Pi_{2}$ be the standard copy of $\Pi$ extending $\Pi^{e}_{2}$ and
containing the edges $e_{x}$, $e_{y}$. Lemma 8.14 applied to the system
$(H,\equiv^{H},\mathscr{H}^{+})$, the copy $\Pi^{e}_{2}$, and the edge $f$
yields $f\in E(\Pi^{e}_{2})$. Thus all copies of $\mathscr{C}$ can be found in
the same standard copy $\Pi_{2}$ and we are in the case discussed immediately
before (9.4).

Second Case. $\mathscr{D}=\Pi^{e}_{1}\overline{W}\Pi^{e}_{2}y$

Here $\Pi^{e}_{1}$ extends to a standard copy $\Pi_{1}$ of $\Pi$ that contains
the edges $f$, $e_{x}$. Moreover, $\overline{W}$ denotes the contraction of
$W$ to $(H,\equiv^{H})$. Due to Lemma 8.11 there exists an edge
$e^{\prime}_{y}$ in $E(\Pi^{e}_{1})\cap E(\overline{W})$ passing through $y$.
By choosing $e^{\prime}_{y}$ instead of $e_{y}$ we again arrive at a situation
where everything relevant happens within the same standard copy, namely
$\Pi_{1}$.

Third Case. $\mathscr{D}=f^{+}x\Pi^{e}_{2}\overline{W}\Pi^{e}_{3}y$ and,
hence, $t=3$.

If $x,y\in V(\overline{W})$, then $\overline{W}\hskip
3.99994pt\scalebox{1.3}[1.3]{{$\blacktriangleleft$}}\hskip 3.99994ptH$ yields
$f\in E(\overline{W})\subseteq E(W)$ and we are done immediately. Now assume
for the sake of contradiction that $\\{x,y\\}\not\subseteq V(\overline{W})$,
which causes $\mathscr{D}$ to be acceptable. Without loss of generality we may
suppose that $\Pi^{e}_{2}$ is a supreme copy of $\mathscr{D}$. The
$\Pi^{e}_{2}$-piece $\Pi^{e}_{3}$ collapses to needs to be short. If
$(e^{\prime}_{y})^{+}$ denotes that piece, then by choosing $e^{\prime}_{y}$
instead of $e_{y}$ we are led to a case with $t=2$, contrary to the minimality
of $t$. ∎

Now we are ready to proceed to the linearity of the pretrain
$(\Sigma,\equiv^{\Sigma})$. Given any two vertices $x$ and $y$, we are to
prove that the set $\mathscr{W}$ of all wagons $W$ with $x,y\in V(W)$ has at
most one element. If there exists an edge $f\in E(\Sigma)$ with $x,y\in f$,
then Claim 9.19 discloses $f\in E(W)$ for every $W\in\mathscr{W}$ and
$|\mathscr{W}|\leq 1$ follows. Thus we may assume that

$\text{ no $f\in E(\Sigma)$ contains both $x$ and $y$}.$ (9.5)

If $x,y\in V(H)$, then the linearity of $(H,\equiv^{H})$ implies that at most
one wagon $W_{\star}^{H}$ of this pretrain satisfies $x,y\in
V(W^{H}_{\star})$. If such a wagon exists, then we denote the unique wagon of
$(\Sigma,\equiv^{\Sigma})$ contracting to it by $W_{\star}$.

Now assume for the sake of contradiction that there exist distinct
$W^{\prime},W^{\prime\prime}\in\mathscr{W}$. In case $W_{\star}$ exists we
suppose, additionally, that $W_{\star}\in\\{W^{\prime},W^{\prime\prime}\\}$.
For every quadruple $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
e\hfil$\crcr}}}=(e^{\prime}_{x},e^{\prime}_{y},e^{\prime\prime}_{x},e^{\prime\prime}_{y})$
of edges with

$x\in e^{\prime}_{x}\in E(W^{\prime})\,,\quad y\in e^{\prime}_{y}\in
E(W^{\prime})\,,\quad x\in e^{\prime\prime}_{x}\in
E(W^{\prime\prime})\,,\quad\text{ and }\quad y\in e^{\prime\prime}_{y}\in
E(W^{\prime\prime})\,,$

the statement (9.5) shows that

$\mathscr{C}=(e^{\prime}_{x})^{+}W^{\prime}(e^{\prime}_{y})^{+}y(e^{\prime\prime}_{y})^{+}W^{\prime\prime}(e^{\prime\prime}_{x})^{+}x$

is a big cycle. As the standard copies of $(\Pi,\equiv^{\Pi})$ are linear
pretrains, there is no such quadruple
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle e\hfil$\crcr}}}$
whose four edges are in the same standard copy. Thus every choice of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle e\hfil$\crcr}}}$
gives rise to at least one segmentation

$\mathscr{C}=I_{1}r_{1}\ldots I_{t}r_{t}$ (9.6)

of the associated cycle $\mathscr{C}$. For the rest of the argument we fix
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle e\hfil$\crcr}}}$
and the segmentation (9.6) in such a manner that $t\in\\{2,3,4\\}$ is minimal.
Let $\mathscr{D}=\Pi^{e}_{1}\overline{r}_{1}\ldots\Pi^{e}_{t}\overline{r}_{t}$
denote the corresponding reduct of $\mathscr{C}$.

First Case. $t=2$

Let us consider the connectors of $\mathscr{D}$. Due to Claim 9.18 and (9.5)
it cannot be the case that both of them are vertices. Moreover, Lemma 8.12
excludes the case that $r_{1}$, $r_{2}$ are the two wagons of $\mathscr{C}$.
Up to symmetry this means that only the possibility
$\mathscr{D}=\Pi^{e}_{1}\overline{W^{\prime}}\Pi^{e}_{2}x$ remains, where
$\Pi^{e}_{1}$ denotes the leader of $(e^{\prime}_{x})^{+}$.

Now Lemma 8.11 provides an edge $f$ such that $x\in f\in
E(\overline{W^{\prime}})\cap E(\Pi^{e}_{2})$ and the quadruple
$(f,e^{\prime}_{y},e^{\prime\prime}_{x},e^{\prime\prime}_{y})$ can play the
rôle of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle e\hfil$\crcr}}}$.
However, all four edges of this quadruple pertain to the same standard copy
$\Pi_{2}$, which is absurd.

Second Case. $t\in\\{3,4\\}$

We contend that $\mathscr{D}$ is acceptable. The property ($A1$) is clear,
($A2$) follows from (9.5), and ($A3$) only requires attention if $t=4$ and the
wagon $W$ we want to test contains both $x$ and $y$. We already know, however,
that there exists at most one such wagon, namely $W^{H}_{\star}$. By our
choice of $W^{\prime}$, $W^{\prime\prime}$ and by $t=4$ this wagon is, if it
exists, a connector of $\mathscr{D}$ and thus irrelevant for ($A3$).

SO $\mathscr{D}$ is indeed acceptable and due to $\mathrm{ord}(\mathscr{D})=2$
it follows that $\mathscr{D}$ has a supreme copy, say $\Pi_{2}^{e}$. If both
segments $I_{1}$, $I_{3}$ contained more than one copy, then the contradiction
$2\leq|\mathscr{C}|-|\mathscr{D}|=4-t\leq 1$ would arise. Consequently we may
assume that $I_{1}$ consists of a single copy, which causes the corresponding
index to be mixed in $\mathscr{D}$. If $P_{1}=f^{+}$ denotes the short
$\Pi_{2}^{e}$-piece $\Pi^{e}_{1}$ collapses to, then by including $f$ into the
quadruple $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle e\hfil$\crcr}}}$
(instead of the edge corresponding to $I_{1}$) we reach a contradiction to the
minimality of $t$.

This concludes the proof that $(\Sigma,\equiv^{\Sigma})$ is linear.

Stage D: The proof of ( ) ‣ 9.2. Consider any acceptable big cycle
$\mathscr{C}=F_{1}q_{1}\ldots F_{n}q_{n}$ in
$(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$ whose order is at most $g$. We are
to prove that $\mathscr{C}$ has a supreme copy.

Assuming that Claim 9.17 does not imply this immediately, we pick a
segmentation

$\mathscr{C}=I_{1}r_{1}\ldots I_{t}r_{t}$

of $\mathscr{C}$ whose length $t\geq 2$ is minimal and pass to the associated
reduct

$\mathscr{D}=\Pi^{e}_{1}\overline{r}_{1}\ldots\Pi^{e}_{t}\overline{r}_{t}\,.$

We commence with the simplest case.

######  Claim 9.20.

If $\mathrm{ord}(\mathscr{D})=1$, then $\mathscr{C}$ has a supreme copy.

###### Proof.

Suppose $\mathscr{D}=\Pi^{e}_{1}x\Pi^{e}_{2}\overline{W}$, where $x$ is a
vertex and $\overline{W}$ is a wagon of $(H,\equiv^{H})$.

Let us first deal with the special case $n=|\mathscr{C}|=2$, where
$\mathscr{C}=F_{1}xF_{2}W$. Due to ($A1$) and symmetry we may assume that the
copy $F_{1}$ is real. Now Lemma 8.11 applied to $\mathscr{D}$ and
$\Pi^{e}_{1}$ yields an edge $f$ such that $x\in f\in E(\overline{W})\cap
E(\Pi^{e}_{1})$. Since all copies of the acceptable big cycle
$\mathscr{E}=F_{1}xf^{+}W$ are in the same standard copy of $\Pi$, Claim 9.17
shows that $\mathscr{E}$ has a supreme copy. Remark 8.8(0 ) tells us that this
supreme copy must be $F_{1}$. The short $F_{1}$-piece $Q$ to which we can
collapse $f^{+}$ witnesses that $F_{1}$ is a supreme copy of $\mathscr{C}$ as
well.

It remains to treat the case $n\geq 3$. If both segments $I_{1}$, $I_{2}$
contained at least two copies of $\mathscr{C}$, then the wagon $W$ would
exemplify that $\mathscr{C}$ violates ($A2$). So we may assume that $I_{1}$
consists of a single copy, say $F_{1}$. This time we utilise Lemma 8.11 for
obtaining an edge $f$ with $x\in f\in E(\overline{W})\cap E(\Pi^{e}_{2})$.
Lemma 8.22( ) ‣ 8.22 tells us that in $\mathscr{C}$ we can replace $F_{1}$ by
$f^{+}$, thus arriving at another acceptable big cycle
$\mathscr{E}=f^{+}xI_{2}W$. The advantage of $\mathscr{E}$ is that all its
copies belong to the same standard copy of $\Pi$. Thus Claim 9.17 tells us
that $\mathscr{E}$ has some supreme copy and owing to Lemma 8.22( ) ‣ 8.22 so
does $\mathscr{C}$. ∎

So in the sequel we can assume
$\mathrm{ord}(\mathscr{D})\in[2,\mathrm{ord}(\mathscr{C})]$. In particular,
$\mathscr{D}$ satisfies ($A1$) and one confirms easily that it inherits the
properties ($A2$), ($A3$) from $\mathscr{C}$. Altogether $\mathscr{D}$ is an
acceptable big cycle whose order is at most $g$; thus

$\mathscr{D}\text{ has a supreme copy.}$ (9.7)

Next, we isolate the following special case.

######  Claim 9.21.

If $t=2$, then $\mathscr{C}$ has a supreme copy.

###### Proof.

As a consequence of Lemma 8.12 it cannot be the case that both connectors of
$\mathscr{D}$ are wagons. So Claim 9.20 allows us to assume that $r_{1}$,
$r_{2}$ are vertices. We distinguish several cases depending on whether the
segments $I_{1}$, $I_{2}$ consist of single copies. Up to symmetry there are
three possibilities.

First Case. $n=2$

Claim 9.16 leads to an edge $f\in E(F_{1})\cap E(F_{2})\cap E(H)$. By ($B1$)
it cannot be the case that both copies of $\mathscr{C}$ are equal to $f^{+}$
and, hence, we may suppose that $F_{1}\neq f^{+}$. So we can collapse $F_{2}$
to the $F_{1}$-piece $f^{+}$, thus inferring that $F_{1}$ is a supreme copy of
$\mathscr{C}$.

Second Case. $n\geq 3$ and $I_{2}$ consists of a singly copy.

To simplify the notation we assume $I_{2}=F_{n}$. Due to Claim 9.16 there
exists an edge $f\in E(F_{n})\cap E(H)$. Now Lemma 8.22( ) ‣ 8.22 informs us
that $\mathscr{E}=I_{1}r_{1}f^{+}r_{2}$ is an acceptable big cycle in
$(\Sigma,\equiv^{\Sigma},\mathscr{Q}^{+})$. Moreover, $r_{1},r_{2}\in f\cap
V(\Pi^{e}_{1})$ and Lemma 8.14 imply $f\in E(\Pi^{e}_{1})$, for which reason
all copies of $\mathscr{E}$ belong to a common standard copy of $\Pi$. By
Claim 9.17 it follows that $\mathscr{E}$ has a supreme copy and, finally, by
Lemma 8.22( ) ‣ 8.22 so does $\mathscr{C}$.

Third Case. Both $I_{1}$, $I_{2}$ contain at least two copies.

Suppose for the sake of contradiction that some edge $f$ of $\Sigma$ covers
$\\{r_{1},r_{2}\\}$. Let $W$ be the wagon of $(\Sigma,\equiv^{\Sigma})$
containing $f$. Due to ($A2$) we know that $W$ cannot serve as a connector of
$\mathscr{C}$ and thus the description of the present case entails that $W$
violates ($A3$). This proves that

$\text{ there is no $f\in E(\Sigma)$ such that $r_{1},r_{2}\in f$}.$ (9.8)

In particular, the copies $\Pi^{e}_{1}$ and $\Pi^{e}_{2}$ need to be real and
due to Lemma 8.13 both of them are supreme copies of $\mathscr{D}$. By Claim
9.18 and symmetry we may suppose that $I_{1}$ does not consist of two copies
and an interposed wagon. Let $P$ be a $\Pi^{e}_{1}$-piece with the property
that collapsing $\Pi^{e}_{2}$ to $P$ establishes the supremacy of
$\Pi_{e}^{1}$ in $\mathscr{D}$. Because of (9.8) we know that $P$ is long.
Write $P=(f^{\prime})^{+}\overline{W}(f^{\prime\prime})^{+}$ and denote the
wagon of $(\Sigma,\equiv^{\Sigma})$ contracting to $\overline{W}$ by $W$.
Owing to ($A3$) this wagon needs to appear in $\mathscr{C}$. Now ($A2$) tells
us that the connectors of $\mathscr{C}$ next to $W$ are $r_{1}$ and $r_{2}$.

So our special choice of $I_{1}$ guarantees that $I_{2}$ consists of two
copies and the wagon $W$ between them. Without loss of generality we may
suppose that $I_{2}=F_{n-1}WF_{n}$. Now Lemma 8.22 applies to $K=\\{n-1,n\\}$,
$P_{n-1}=(f^{\prime})^{+}$, and $P_{n}=(f^{\prime\prime})^{+}$. So by part ( )
‣ 8.22 of the lemma $I_{1}r_{1}(f^{\prime})^{+}W(f^{\prime\prime})^{+}r_{2}$
is an acceptable big cycle, by Claim 9.17 this cycle has a supreme copy, and
by Lemma 8.22( ) ‣ 8.22 $\mathscr{C}$ has a supreme copy as well. ∎

Throughout the remainder of this proof we suppose that $n\geq t\geq 3$. Let
$\Pi^{e}_{\star}$ be a supreme copy of $\mathscr{D}$ (see (9.7)) and let

$\bigl{\\{}Q_{\tau}\colon\tau\in{\mathds{Z}}/t{\mathds{Z}}\text{ and
}\Pi_{\tau}^{e}\neq\Pi_{\star}^{e}\bigr{\\}}$

be a family of $\Pi^{e}_{\star}$-pieces acting as supremacy witnesses. Our
goal is to associate with every copy $F_{k}$ of $\mathscr{C}$ that belongs to
a segment $I_{\tau}$ whose leader $\Pi_{\tau}^{e}$ is distinct from
$\Pi_{\star}^{e}$ a $\Pi^{e}_{\star}$-piece $P_{k}$ such that

  1. (0 )

if one eliminates $F_{k}$ from $\mathscr{C}$ and inserts $P_{k}$ instead of
it, then the resulting cyclic sequence satisfies ($B3$) and ($B4$),

  2. (0 )

and if $P_{k}$ is long, then it has the properties ( ) ‣ 8.22, ( ) ‣ 8.22, and
( ) ‣ 8.22 from Lemma 8.22.

Once we have such pieces $P_{k}$ at our disposal, we can perform all
replacements indicated in (0 ) simultaneously and Lemma 8.22( ) ‣ 8.22 informs
us that the cyclic sequence that results is an acceptable big cycle. All
copies of this cycle will be contained in the same standard copy
$(\Pi_{\star},\equiv^{\Pi_{\star}},\mathscr{P}_{\star}^{+})$, where
$\Pi_{\star}$ extends $\Pi^{e}_{\star}$. Owing to Claim 9.17 the modified
cycle needs to have a supreme copy, so by Lemma 8.22( ) ‣ 8.22 the original
big cycle $\mathscr{C}$ has a supreme copy as well.

Thus it suffices to exhibit the $\Pi^{e}_{\star}$-pieces described above.
Notice that the connectors $\overline{r}_{\tau-1}$ and $\overline{r}_{\tau}$
cannot be wagons at the same time, for then the piece $Q_{\tau}$ could not
exist. Suppose next that one of those connectors, say $\overline{r}_{\tau-1}$,
is a vertex while the other one, i.e. $\overline{r}_{\tau}$, is a wagon. In
this case the piece $Q_{\tau}$ needs to be short. Besides, since the wagon
$r_{\tau}$ satisfies ($A2$) in $\mathscr{C}$, the segment $I_{\tau}$ has
length $1$ and we can set $P_{k}=Q_{\tau}$.

The last case we need to consider is that both $r_{\tau-1}$ and $r_{\tau}$ are
vertices. Suppose first that the piece $Q_{\tau}=f^{+}$ is short and let $W$
be the wagon of $(\Sigma,\equiv^{\Sigma})$ with $f\in E(W)$. An easy argument
using ($A2$) shows that $W$ cannot be a connector of $\mathscr{C}$. So ($A3$)
applies to $W$. We deduce that the segment $I_{\tau}$ has length $1$ and again
we can set $P_{k}=Q_{\tau}$.

So henceforth we may assume that the piece
$Q_{\tau}=(f^{\prime})^{+}\overline{W}(f^{\prime\prime})^{+}$ is long. The
easiest possibility would be that the interval $I_{\tau}$ is of the form
$F_{j}WF_{j+1}$, where $W$ denotes the wagon of $(\Sigma,\equiv^{\Sigma})$
contracting to $\overline{W}$, for then we just need to set
$P_{j}=(f^{\prime})^{+}$ and $P_{j+1}=(f^{\prime\prime})^{+}$.

Let us now assume for the sake of contradiction that $I_{\tau}$ is not of this
special form. Observe first that the admissibility of $Q_{\tau}$ as a
supremacy witness entails that

no edge of $H$ contains both $r_{\tau-1}$ and $r_{\tau}$. (9.9)

If the connectors $r_{\tau-1}$ and $r_{\tau}$ are consecutive in
$\mathscr{C}$, then Claim 9.16 applied to the copy between them yields an edge
that contradicts (9.9). This shows that

$\text{the connectors $r_{\tau-1}$ and $r_{\tau}$ fail to be consecutive in
$\mathscr{C}$}.$ (9.10)

Now condition ($A3$) tells us that the wagon $W$ has to appear somewhere in
$\mathscr{C}$. On the other hand, $\overline{W}$ cannot appear in
$\mathscr{D}$, for then the piece $Q_{\tau}$ would again not be admissible. In
other words, $W$ is hidden in one of the segments. But by ($A2$) and $t\geq 3$
this is only possible if $I_{\tau}$ is indeed of the desired form. ∎

The following should now be straightforward, but let us elaborate.

###### Proof of Proposition 9.14.

Consider a pretrain $(F,\equiv^{F})$ with $\mathfrak{girth}(F,\equiv^{F})>g$
and a number of colours $r$. Construct the linear system
$\Phi_{r}(F)=(G,\mathscr{G})$ with strongly induced copies and enumerate the
edges of $G$ arbitrarily as $E(G)=\\{e(1),\ldots,e(N)\\}$. By Lemma 8.17 the
picture zero $(\Pi_{0},\equiv_{0},\mathscr{P}_{0},\psi_{0})$ corresponding to
this situation satisfies
${\mathfrak{Girth}(\Pi_{0},\equiv_{0},\mathscr{P}_{0}^{+})>g}$.

Starting with picture zero we intend to execute a partite construction, i.e.,
to construct recursively a sequence of pictures
$(\Pi_{\alpha},\equiv_{\alpha},\mathscr{P}_{\alpha},\psi_{\alpha})_{\alpha\leq
N}$ over $(G,\mathscr{G})$. Suppose that for some $\alpha\in[N]$ we have just
constructed a picture
$(\Pi_{\alpha-1},\equiv_{\alpha-1},\mathscr{P}_{\alpha-1},\psi_{\alpha-1})$
with

$\mathfrak{Girth}(\Pi_{\alpha-1},\equiv_{\alpha-1},\mathscr{P}_{\alpha-1}^{+})>g\,.$

In particular, we have
$\mathfrak{Girth}\bigl{(}\Pi_{\alpha-1},\equiv_{\alpha-1},E^{+}(\Pi_{\alpha-1})\bigr{)}>g$
and Lemma 8.17 reveals

$\mathfrak{girth}(\Pi_{\alpha-1},\equiv_{\alpha-1})>g\,.$

This in turn implies that $\Xi_{r}(\cdot)$ applies to the constituent
$\bigl{(}\Pi_{\alpha-1}^{e(\alpha)},\equiv_{\alpha-1}^{e(\alpha)}\bigr{)}$,
where $\equiv_{\alpha-1}^{e(\alpha)}$ denotes the restriction of
$\equiv_{\alpha-1}$ to $E\bigl{(}\Pi_{\alpha-1}^{e(\alpha)}\bigr{)}$. In other
words,

$\Xi_{r}\bigl{(}\Pi_{\alpha-1}^{e(\alpha)},\equiv_{\alpha-1}^{e(\alpha)}\bigr{)}=(H_{\alpha},\equiv^{H_{\alpha}},\mathscr{H}_{\alpha})$

is defined and satisfies
$\mathfrak{Girth}(H_{\alpha},\equiv^{H_{\alpha}},\mathscr{H}_{\alpha}^{+})>g$.
By Lemma 9.15 the new picture

$(\Pi_{\alpha},\equiv_{\alpha},\mathscr{P}_{\alpha},\psi_{\alpha})=(\Pi_{\alpha-1},\equiv_{\alpha-1},\mathscr{P}_{\alpha-1}\psi_{\alpha-1})\mathbin{\mathchoice{\leavevmode\vtop{
\halign{\hfil$\m@th\displaystyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\textstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptstyle#$\hfil\cr+\cr\times\crcr}}}{\leavevmode\vtop{
\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr+\cr\times\crcr}}}}(H_{\alpha},\equiv^{H_{\alpha}},\mathscr{H}_{\alpha})$

has the property
$\mathfrak{Girth}(\Pi_{\alpha},\equiv_{\alpha},\mathscr{P}_{\alpha}^{+})>g$
and thus the partite construction goes on.

Eventually we obtain the final picture
$\mathrm{PC}(\Phi,\Xi)_{r}(F,\equiv^{F})=(\Pi_{N},\equiv_{N},\mathscr{P}_{N})$
with

$\mathfrak{Girth}(\Pi_{N},\equiv_{N},\mathscr{P}_{N}^{+})>g\,.\qed$

###  9.3. Orientation

The main results of this section lead to a quick solution of the problem
discussed in §7.5. The construction
$\varUpsilon=\mathrm{Ext}(\Omega^{(2)},\Omega^{(2)})$ applies to all linear,
ordered, $f$-partite pretrains. Due to the case $g=2$ of Lemma 9.1 the systems
of pretrains $(H,\equiv^{H},\mathscr{H})$ generated by $\varUpsilon$ satisfy
$\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>2$ (the hypotheses of this
lemma were verified in Proposition 3.8 and Corollary 5.6).

Now we can use $\varUpsilon$ as a partite lemma and look at the construction
$\Lambda=\mathrm{PC}(\Omega^{(2)},\varUpsilon)$. The case $g=2$ of Proposition
9.14 informs us that for every linear, ordered, $f$-partite pretrain
$(F,\equiv^{F})$ and every number of colours $r$ the system of pretrains
$\Lambda_{r}(F,\equiv^{F})=(H,\equiv^{H},\mathscr{H})$ is defined and
satisfies $\mathfrak{Girth}(H,\equiv^{H},\mathscr{H}^{+})>2$. So, in
particular, $(H,\equiv^{H})$ is a linear pretrain arrowing $(F,\equiv^{F})$
with $r$ colours. Moreover, and this is the main advantage of $\Lambda$ over
$\varUpsilon$, the copies in $\mathscr{H}$ have clean intersections due to
Lemma 3.4.

##  10\. Trains

Let us start this section with an observation of central importance to our
girth Ramsey theoretic endeavour: a “typical” $k$-partite $k$-uniform
hypergraph can never occur as a constituent of a picture that arises when the
partite construction method is carried out over a linear system
$(G,\mathscr{G})$.

For instance, all constituents of picture zero consist of a matching together
with some isolated vertices (cf. Figure 3.2). The first picture is obtained by
applying the intended partite lemma to the constituent over some edge $e(1)\in
E(G)$ and performing the usual amalgamation. Evidently, every constituent of
the new picture over an edge $e\in E(G)$ with $e(1)\cap e=\varnothing$ is
still a matching augmented by some isolated vertices. More interestingly, if
an edge $e\neq e(1)$ intersects $e(1)$, then by the linearity of $G$ this
intersection occurs in a single vertex $x$ and, accordingly, the new
constituent over $e$ is obtained by taking several disjoint copies of the old
constituent and identifying a couple of vertices on the music line $V_{x}$.
However, there occur no such identifications on the other music lines $V_{z}$
with $z\in e\smallsetminus\\{x\\}$. Notice that we can endow the new
constituent with a pretrain structure by declaring two edges to be equivalent
if they belong to the same copy of the old constituent. It should also be
clear that this pretrain is going to be linear if our partite lemma delivers
copies with clean intersections.

Similarly, the most complicated possibility for a constituent of the second
picture is that one takes plenty of disjoint copies of such a pretrain and
identifies some of their vertices, all identified vertices belonging to a
common music line. As the partite construction progresses, there iteratively
arise more and more complex objects all of which fit into the framework we
shall now develop (see also Figure 2.3).

###  10.1. Quasitrains and parameters

We begin by encoding the recursive formation of our objects in terms of a
sequence of nested equivalence relations.

######  Definition 10.1.

Given a positive integer $m$ we say that
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0},\ldots,\equiv_{m})$ is a quasitrain of height
$m$ if $F$ is a hypergraph and $\equiv_{0},\ldots,\equiv_{m}$ are equivalence
relations on $E(F)$ such that the following holds.

  1. ( )

Every wagon of the pretrain $(F,\equiv_{0})$ consists of a single edge.

  2. ( )

If $e^{\prime},e^{\prime\prime}\in E(F)$, $\mu\in[0,m)$, and
$e^{\prime}\equiv_{\mu}e^{\prime\prime}$, then
$e^{\prime}\equiv_{\mu+1}e^{\prime\prime}$.

  3. ( )

Any two edges of $F$ are equivalent with respect to $\equiv_{m}$.

For every $\mu\in[0,m]$ the wagons of the pretrain $(F,\equiv_{\mu})$ are
called the $\mu$-wagons of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.

######  Example 10.2.

Quasitrains of height $1$ are triples $(F,\equiv_{0},\equiv_{1})$, where $F$
is a hypergraph, the $0$-wagons are single edges (by ( ) ‣ 10.1), and all
edges belong to the same $1$-wagon (by ( ) ‣ 10.1). Consequently, there is a
natural bijective correspondence between quasitrains of height $1$ and
hypergraphs. At a later moment it will be convenient to call
$(F,\equiv_{0},\equiv_{1})$ the quasitrain of height $1$ associated with the
hypergraph $F$.

######  Example 10.3.

Similarly, quasitrains of height $2$ are quadruples
$(F,\equiv_{0},\equiv_{1},\equiv_{2})$ with the properties that
$(F,\equiv_{0},\equiv_{2})$ is a quasitrain of height $1$ and $(F,\equiv_{1})$
is a pretrain. In particular, quasitrains of height $2$ are in natural
bijective correspondence with pretrains.

The train concept to which we proceed next is taking into account what we said
earlier about amalgamations occurring only in prespecified music lines. Recall
that in Definition 3.15( ) ‣ 3.15 we introduced the following convenient piece
of notation for such situations: given an $f$-partite hypergraph $F$ with
index set $I$ and a subset $A\subseteq I$ the union $\bigcup_{i\in A}V_{i}(F)$
of the vertex classes whose indices belong to $A$ is denoted by $V_{A}(F)$.

######  Definition 10.4.

Suppose that $m\in{\mathds{N}}$, that $f\colon I\longrightarrow{\mathds{N}}$
is a function from a finite index set $I$ to ${\mathds{N}}$ with $\sum_{i\in
I}f(i)\geq 2$, and that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
A\hfil$\crcr}}}=(A_{1},\ldots,A_{m})\in\powerset(I)^{m}$ is a sequence of
subsets of $I$. A structure $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0},\ldots,\equiv_{m})$ is said to be an $f$-partite
train of height $m$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
(see Figure 10.1) if it has the following two properties.

  1. ( )

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
an $f$-partite quasitrain of height $m$.

  2. ( )

If $\mu\in[m]$ and $W^{\prime}_{\mu-1}$, $W^{\prime\prime}_{\mu-1}$ are two
distinct $(\mu-1)$-wagons of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
included in a common $\mu$-wagon, then $V(W^{\prime}_{\mu-1})\cap
V(W^{\prime\prime}_{\mu-1})\subseteq V_{A_{\mu}}(F)$.

######  Example 10.5.

Trains of height $1$ have one-term sequences serving as their parameters. If
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0},\equiv_{1})$ is an $f$-partite quasitrain of
height $1$ (cf. Example 10.2), $I$ denotes the domain of $f$, and
$A_{1}\subseteq I$, then $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
a train with parameter $(A_{1})$ if and only if $e^{\prime}\cap
e^{\prime\prime}\subseteq V_{A_{1}}(F)$ holds for any two distinct edges
$e^{\prime},e^{\prime\prime}\in E(F)$, i.e., if $F$ is $A_{1}$-intersecting in
the sense of Definition 3.15( ) ‣ 3.15.

$W^{\prime}$$W^{\prime\prime}$$W^{\prime\prime\prime}$$1$$2$$3$ Figure 10.1.
An $f$-partite train of height $1$ with parameter $(A_{1})$, where
$f\colon[3]\longrightarrow{\mathds{N}}$ is defined by $f(1)=f(3)=1$, $f(2)=2$,
and $A_{1}=\\{1,2\\}$.

When compared to the discussion at the beginning of this section, there are
two directions in which our definition of trains might appear to be too
general. First, we only motivated the need for $k$-partite $k$-uniform trains,
whereas Definition 10.4 allows the underlying hypergraphs of trains to be
$f$-partite for general functions $f$. The reason for this is that we intend
to subject trains to the extension process and, as we saw in Section 6, when
we apply a construction of the form $\mathrm{Ext}(\Phi,\Psi)$ to a $k$-partite
$k$-uniform pretrain, the construction $\Psi$ needs to be applicable to
general $f$-partite hypergraphs.

Another surprise in Definition 10.4 might be that we allow the sets
$A_{1},\dots,A_{m}$ constituting the parameter of the train to be arbitrary
subsets of $I$. After all, we have just tried to convey the idea that for
trains created by partite constructions we can demand $|A_{\mu}|\leq 1$ for
every $\mu\in[m]$, and that this very fact will be responsible for the girth
increment the achievement of which is the reason for starting to consider
trains at all. On the other hand, when we have some construction applicable to
$f$-partite trains and attempt to clean it by means of the partite
construction method, then from the point of view of the constituents of the
involved pictures the sizes of the sets in the parameter can appear to have
been enlarged. This tension between conflicting demands on the parameters will
completely resolve itself in Section 12.

It is probably clear what we mean by subtrains, but let us elaborate for the
sake of completeness.

######  Definition 10.6.

Given two quasitrains

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0}^{F},\ldots,\equiv_{m}^{F})\quad\text{ and
}\quad\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}}=(G,\equiv_{0}^{G},\ldots,\equiv_{m}^{G})$

of the same height $m$ we say that
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
a subquasitrain of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle G\hfil$\crcr}}}$ if

  1. ( )

$F$ is a subhypergraph of $G$

  2. ( )

and $\forall e^{\prime},e^{\prime\prime}\in
E(F)\,\,\forall\mu\in[0,m]\,\,\bigl{[}e^{\prime}\equiv^{F}_{\mu}e^{\prime\prime}\,\,\,\Longleftrightarrow\,\,\,e^{\prime}\equiv^{G}_{\mu}e^{\prime\prime}\bigr{]}$.

If, moreover, both $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
and $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle G\hfil$\crcr}}}$
are $f$-partite trains having the same parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
and ( ) ‣ 10.6 holds in the stronger form that $F$ is an $f$-partite
subhypergraph of $G$, then $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
called a subtrain of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle G\hfil$\crcr}}}$.

The subtrain relation is, of course, reflexive and transitive. A quasitrain
system of height $m$ is a structure of the form

$(H,\equiv_{0}^{H},\ldots,\equiv_{m}^{H},\mathscr{H})\,,$

where $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv_{0}^{H},\ldots,\equiv_{m}^{H})$ is a quasitrain of
height $m$ and $\mathscr{H}$ is a collection of subquasitrains of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$.
It will be convenient to denote this system by
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ as well. We regard the notion of an extended
quasitrain system $(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})$ to be self-explanatory.

###  10.2. More German girth

Next we need to generalise $\mathfrak{girth}$ to quasitrains, and
$\mathfrak{Girth}$ to systems of quasitrains. In both contexts we work with
the free monoid $\mathfrak{M}$ generated by ${\mathds{N}}_{\geq 2}$. Thus the
elements of $\mathfrak{M}$ can be thought of as sequences of elements from
${\mathds{N}}_{\geq 2}$ and the composition of $\mathfrak{M}$, denoted by
$\circ$, is the concatenation of sequences. The empty sequence $\varnothing$
is the neutral element of $\mathfrak{M}$. For a sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}=(g_{1},\dots,g_{m})\in\mathfrak{M}$ we call
$m=|\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}|$
the length of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and we use $\inf(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}})$
as an abbreviation for $\inf\\{g_{1},\dots,g_{m}\\}$. So we have
$\inf(\varnothing)=\infty$ and in all other cases the infimum is just a
minimum. For integers $g\geq 2$ and $m\geq 1$ the sequence $(g,\dots,g)$
consisting of $m$ terms equal to $g$ is the $m^{\mathrm{th}}$ power of the
one-term sequence $(g)$ and thus we shall denote it by $(g)^{m}$. Finally, we
write $\mathfrak{M}_{\leq}$ for the subset of $\mathfrak{M}$ consisting of all
nondecreasing sequences (including $\varnothing$), and we set
$\mathfrak{M}^{\times}_{\leq}=\mathfrak{M}_{\leq}\smallsetminus\\{\varnothing\\}$.

A straightforward adaptation of Definition 8.16 leads to the following
$\mathfrak{girth}$ notion for quasitrains.

######  Definition 10.7.

Given a quasitrain $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0},\ldots,\equiv_{m})$ of height $m$ and a sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}=(g_{1},\dots,g_{m})\in\mathfrak{M}$ of length $m$ we write
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$ if
for every $\mu\in[m]$ and every $\mu$-wagon $W_{\mu}$ we have

$\mathfrak{girth}(W_{\mu},\equiv_{\mu-1}^{W_{\mu}})>g_{\mu}\,,$

where $\equiv^{W_{\mu}}_{\mu-1}$ denotes the restriction of $\equiv_{\mu-1}$
to $E(W_{\mu})$. A quasitrain $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ of
height $m$ with
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>(2)^{m}$ is said to be linear.

The transitivity of ordinary girth leads to the following statement.

######  Lemma 10.8.

Let $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}=(g_{1},\dots,g_{m})\in\mathfrak{M}^{\times}_{\leq}$ be given.
If $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0}^{F},\ldots,\equiv_{m}^{F})$ denotes a quasitrain
of height $m$ with
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$,
then $\mathfrak{girth}(F,\equiv_{\mu-1})>g_{\mu}$ holds for every $\mu\in[m]$.
In particular, we have $\mathrm{girth}(F)>g_{1}$.

###### Proof.

The case $\mu=m$ is clear, for by Definition 10.1( ) ‣ 10.1 there is only one
$m$-wagon of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
and by Definition 10.7 applied to that wagon we obtain indeed
$\mathfrak{girth}(F,\equiv_{m-1})>g_{m}$.

Arguing by decreasing induction on $\mu$ we now assume that some $\mu\in[m-1]$
has the property $\mathfrak{girth}(F,\equiv_{\mu})>g_{\mu+1}\geq g_{\mu}$,
meaning that the $\mu$-wagons of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
form a set system whose girth exceeds $g_{\mu}$. According to Definition 10.7
each of these $\mu$-wagons is composed of $(\mu-1)$-wagons the vertex sets of
which form a set system whose girth is larger than $g_{\mu}$ as well. Due to
Fact 4.2 this implies $\mathfrak{girth}(F,\equiv_{\mu-1})>g_{\mu}$ and the
induction is complete.

As the $0$-wagons consist of single edges, the special case $\mu=1$ yields
$\mathrm{girth}(F)>g_{1}$. ∎

The reason why we are keeping track of suitable parameters when discussing
trains is that they are needed for the following important yet innocent
looking variant of this argument.

######  Lemma 10.9.

Let $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv_{0},\ldots,\equiv_{m})$ be a $k$-partite $k$-uniform
train whose parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
B\hfil$\crcr}}}=(B_{1},\dots,B_{m})$ satisfies $|B_{\mu}|\leq 1$ for every
$\mu\in[m]$. If
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>(g)^{m}$ holds for some integer $g\geq 2$, then
$\mathrm{girth}(F)>g+1$.

###### Proof.

We are to prove that the girth of the unique $m$-wagon of $F$ exceeds $g+1$.
Arguing indirectly, we let $\mu\in[0,m]$ be minimal with the property that for
some $\mu$-wagon $W_{\mu}$ of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
the statement $\mathrm{girth}(W_{\mu})>g+1$ fails.

As the $0$-wagons of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
are single edges, we have $\mu>0$. Therefore $W_{\mu}$ is comprised of
$(\mu-1)$-wagons and by the minimality of $\mu$ the girth of each of them is
larger than $g+1$. According to Definition 10.7 the vertex sets of these
$(\mu-1)$-wagons form a set system whose girth is larger than $g$. Due to Fact
4.2 it follows that $W_{\mu}$ contains a $(g+1)$-cycle $e_{1}v_{1}\ldots
e_{g+1}v_{g+1}$ and distinct $(\mu-1)$-wagons
$W^{1}_{\mu-1},\ldots,W^{g+1}_{\mu-1}$ with $e_{i}\subseteq V(W^{i}_{\mu-1})$
for every $i\in{\mathds{Z}}/(g+1){\mathds{Z}}$. Now clause ( ) ‣ 10.4 of
Definition 10.4 implies $v_{1},\ldots,v_{g+1}\in V_{B_{\mu}}$. Together with
$|B_{\mu}|\leq 1$ this shows, in particular, that $v_{1}$, $v_{g+1}$ are in
the same vertex class of $F$. On the other hand, these two vertices are
distinct and belong to the edge $e_{1}$. We have thereby reached a
contradiction to the assumption that $F$ be $k$-partite and $k$-uniform for
some $k\geq 2$, and the proof is complete. ∎

######  Definition 10.10.

Given an extended quasitrain system
$(H,\equiv_{0},\ldots,\equiv_{m},\mathscr{H}^{+})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})$ of height $m$, and a sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}=(g_{\ell},\dots,g_{m})\in\mathfrak{M}$ whose length is at most
$m$ we write $\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$ if

  1. $\bullet$

for every $\mu\in[\ell,m]$ we have
$\mathfrak{Girth}(H,\equiv_{\mu-1},\mathscr{H}^{+})>g_{\mu}$,

  2. $\bullet$

and $\mathfrak{Girth}(H,\equiv_{m},\mathscr{H}^{+})>1$.

In practice we will always have $\ell\in\\{1,2\\}$ when working with this
definition. The case $\ell=1$ is certainly more natural, as it corresponds to
having a demand on $\mathfrak{Girth}(H,\equiv_{\mu-1},\mathscr{H}^{+})$ for
every possible value of $\mu$. If a train system
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ is created by means of the extension process,
however, there will be copies intersecting in entire $1$-wagons (see §11.2)
and thus there is nothing interesting that could be said about
$\mathfrak{Girth}(H,\equiv_{0},\mathscr{H}^{+})$. This lack of information is
the reason why we sometimes have to resort to the case $\ell=2$. Fortunately,
the partite construction method allows us to regain control over
$\mathfrak{Girth}(H,\equiv_{0},\mathscr{H}^{+})$ and thus we can prevent this
deficit from spreading (see §12.4). So we never need to deal with the
possibility $\ell>2$. Let us end this discussion by recording a direct
consequence of Lemma 8.17.

######  Corollary 10.11.

If a sequence $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}$ and a quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv_{0},\ldots,\equiv_{m})$ of height
$m=|\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}|$
satisfy
$\mathfrak{Girth}\bigl{(}\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},E^{+}(H)\bigr{)}>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$,
then $\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
holds as well. ∎

###  10.3. Diamonds

We shall now give a precise description of the induction scheme the proof of
the girth Ramsey theorem is based on. The central statement concerns trains of
large $\mathfrak{girth}$ and reads as follows.

######  Definition 10.12.

For every sequence $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}_{\leq}^{\times}$ of length $m$ the Ramsey
theoretic principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
asserts that there exists a construction
$\Psi^{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
associating with every ordered $f$-partite train

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$

of height $m$ satisfying
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and with every number of colours $r$ an ordered $f$-partite train system

$\Psi^{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}}_{r}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m},\mathscr{H})=(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$

with the same parameter as $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
such that the copies in $\mathscr{H}$ are strongly induced,

$\mathscr{H}\longrightarrow(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})_{r}\,,\quad\text{ and
}\quad\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}\,.$

Essentially we shall prove all these principles “by induction on
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$”.
Up to some changes in the language, the base case has already been analysed.

######  Lemma 10.13.

The principle $\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(2)}$ holds.

###### Proof.

Let an ordered $f$-partite train $(F,\equiv^{F}_{0},\equiv^{F}_{1})$ of height
$1$ with parameter $(A)$ and a number of colours $r$ be given. For reasons of
simplicity we assume first that $F$ has no isolated vertices.

Now we form the hypergraph system $\Omega^{(2)}_{r}(F)=(H,\mathscr{H})$, and
consider the quasitrain $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\equiv^{H}_{1})$ associated with $H$ (as in
Example 10.2). Corollary 5.6 informs us that $H$ is an ordered $f$-partite
hypergraph, $\mathscr{H}\longrightarrow(F)_{r}$, and
$\mathrm{Girth}(H,\mathscr{H}^{+})>2$. Since $F$ is $A$-intersecting (cf.
Example 10.5), Lemma 3.18 shows that $H$ is $A$-intersecting as well or, in
other words, that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ is
a train with parameter $(A)$.

By Proposition 3.8 the copies in $\mathscr{H}$ are strongly induced in $H$.
Moreover, Lemma 8.18 yields
$\mathfrak{Girth}(H,\equiv^{H}_{0},\mathscr{H}^{+})>2$. Since $F$ has no
isolated vertices, Lemma 8.11 implies
$\mathfrak{Girth}(H,\equiv^{H}_{1},\mathscr{H}^{+})>1$. According to
Definition 10.10 this shows
${\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>(2)}$ and altogether the system
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ is as demanded by
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(2)}$. This concludes our discussion
of the case that $F$ has no isolated vertices.

In the general case we remove the isolated vertices from $F$, perform the
construction we have just seen, and then we put the isolated vertices back.
This needs to be done in such a way that each copy in $\mathscr{H}$ ends up
getting its own isolated vertices, so that
$\mathfrak{Girth}(H,\equiv^{H}_{1},\mathscr{H}^{+})>1$ remains valid. ∎

There will be two quite distinct kinds of induction steps for traversing
$\mathfrak{M}_{\leq}^{\times}$. First, the extension process together with
some subsequent cleaning steps will allow us to prove the following
implication.

######  Proposition 10.14.

For every nonempty nondecreasing sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}_{\leq}^{\times}$ the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
implies
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(2)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.

Second, we shall later describe a diagonal variant of the partite construction
method and establish the following result.

######  Proposition 10.15.

Suppose that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}_{\leq}$ is a (possibly empty) nondecreasing
sequence and that $g$ is an integer satisfying
$\inf(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})>g\geq 2$. If for every positive integer $m$ the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
holds, then
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
is likewise valid.

Throughout the remainder of this subsection, we assume that these two
propositions are true and explore some of their consequences. In particular,
we show that they really yield all karo principles.

######  Lemma 10.16.

For every sequence $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}_{\leq}^{\times}$ and every integer $g$ such
that $\inf(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})\geq g\geq 2$ the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
implies
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.

###### Proof.

We argue by induction on $g$. Proposition 10.14 provides the base case $g=2$.
Now suppose that the lemma holds for some integer $g\geq 2$ and that some
nonempty nondecreasing sequence
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}_{\leq}^{\times}$ satisfying
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
and $\inf(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})\geq g+1$ is given. Iterative applications of the induction
hypothesis establish
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
for every positive integer $m$ and thus Proposition 10.15 yields the desired
principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$.
∎

######  Lemma 10.17.

For every integer $g\geq 2$ the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)}$ holds.

###### Proof.

Referring to Lemma 10.13 as a base case we argue by induction on $g$. In the
induction step we suppose that $g\geq 2$ denotes an integer such that
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)}$ is true. Repeated applications
of the previous lemma disclose
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)^{m}}$ for every positive integer
$m$ and appealing once more to Proposition 10.15, this time with the empty
sequence, we infer $\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g+1)}$. ∎

It should be clear that the two foregoing lemmata suffice for proving the
following statement by induction on
$|\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}|$.

######  Corollary 10.18.

For every sequence $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}_{\leq}^{\times}$ the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}}$
is valid. ∎

Moreover, when unravelling the meaning of Lemma 10.17 one arrives at a fairly
strong form of the girth Ramsey theorem first announced in §4.4.

######  Theorem 10.19.

For every integer $g\geq 2$ there exists a Ramsey construction $\Omega^{(g)}$
that given an ordered $f$-partite hypergraph $F$ with $\mathrm{girth}(F)>g$
and a number of colours $r$ produces an ordered $f$-partite system
$\Omega^{(g)}_{r}(F)=(H,\mathscr{H})$ satisfying
$\mathscr{H}\longrightarrow(F)_{r}$ and $\mathrm{Girth}(H,\mathscr{H}^{+})>g$.

###### Proof.

Let $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\equiv^{F}_{1})$ be the ordered $f$-partite
quasitrain associated to $F$ (as in Example 10.2). We can view
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ as
a train with parameter $(I)$, where $I$ denotes the domain of $f$. Since
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>(g)$, the principle
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{(g)}$ yields a train system
$(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H})$ such that
$\mathscr{H}\longrightarrow(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})_{r}$ and
$\mathfrak{Girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}},\mathscr{H}^{+})>(g)$. Writing
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\equiv^{H}_{1})$ it remains to observe that
$\mathfrak{Girth}(H,\equiv^{H}_{0},\mathscr{H}^{+})>g$ implies
$\mathrm{Girth}(H,\mathscr{H}^{+})>g$ (see Lemma 8.18). ∎

In analogy to Lemma 3.18 we could add that the construction $\Omega^{(g)}$
also preserves being $A$-intersecting, but due to a lack of known applications
we do not state this more carefully here. Let us finally recall that every
$k$-uniform hypergraph $F$ can be viewed as an $f$-partite hypergraph, where
$f$ denotes any function from a one-element set to $\\{k\\}$. Therefore,
Theorem 10.19 immediately yields a Ramsey theorem for arbitrary hypergraphs
without short cycles. As the conclusion $\mathrm{Girth}(H,\mathscr{H}^{+})>g$
implies $\mathrm{girth}(H)>g$ (see Lemma 4.12) we thereby see that Theorem
10.19 yields Theorem 1.4. Let us collect all these implications into a single
statement.

######  Summary 10.20.

The conjunction of Proposition 10.14 and Proposition 10.15 leads to Corollary
10.18, Theorem 10.19, and to the girth Ramsey theorem as formulated in Theorem
1.4.

######  Remark 10.21.

With the help of ordinal numbers our induction scheme can be reformulated in
a, perhaps, more transparent way. The map
$\varphi\colon\mathfrak{M}_{\leq}\longrightarrow\omega^{\omega}$ defined by

$\varphi(g_{1},\dots,g_{m})=\omega^{g_{m}-2}+\dots+\omega^{g_{1}-2}$

is bijective. Thus Corollary 10.18 asserts that
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\varphi^{-1}(\alpha)}$ holds for every
positive ordinal $\alpha$ beneath $\omega^{\omega}$. The proof is by induction
on $\alpha$, its base case $\alpha=1$ agrees with Lemma 10.13, and Proposition
10.14 takes care of the successor step
$\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\varphi^{-1}(\alpha)}\Longrightarrow\scalebox{1.2}[1.2]{{$\diamondsuit$}}_{\varphi^{-1}(\alpha+1)}$.
Finally, Proposition 10.15 provides the limit step, for in the notation
employed there the sequence
$\langle\varphi((g)^{m}\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}})\colon m<\omega\rangle$ converges to the limit number
$\varphi((g+1)\circ\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}})$.

##  11\. Trains in the extension process

In this section we take the first major step towards proving Proposition
10.14. Notice that we are given there a construction applicable to certain
trains of height $m=|\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}|$,
and we are to exhibit another construction capable of handling certain trains
of height $m+1$. The plan for accomplishing this height increment is to use
the extension process; the main result of this section describes how far we
can go with this idea (see Lemma 11.10). The main deficit of the construction
we obtain is that the copies it produces can still intersect in entire
$1$-wagons. Following the arguments we saw in §9.3 this can be remedied by
means of the partite construction method, but we defer the details of this
cleaning step to §12.4.

###  11.1. Extensions of trains

The extension process discussed in Section 6 generalises from pretrains to
quasitrains of arbitrary height and our next immediate goal is to develop an
appropriate language for the description of such constructions. In other
words, we repeat §6.2 in a more intricate setting.

######  Definition 11.1.

Given a subquasitrain $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ of a quasitrain

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})$

we call $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ a
$1$-extension of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
provided that $(H,\equiv^{H}_{1})$ is an extension of $(F,\equiv^{F}_{1})$.

Let us recall that the latter notion was introduced in Definition 6.2 and in
the present situation it means that the following three conditions hold.

  1. ( )

The hypergraphs $F$ and $H$ have the same isolated vertices.

  2. ( )

Every $1$-wagon $W$ of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
contracts to a $1$-wagon of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$,
i.e., satisfies $E(W)\cap E(F)\neq\varnothing$.

  3. ( )

If two distinct $1$-wagons $W_{1}^{\star}$ and $W_{1}^{\star\star}$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
contract to the $1$-wagons $\overline{W}_{1}^{\star}$ and
$\overline{W}_{1}^{\star\star}$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$,
then $V(W_{1}^{\star})\cap
V(W_{1}^{\star\star})=V(\overline{W}_{1}^{\star})\cap
V(\overline{W}_{1}^{\star\star})$.

The next lemma expresses the intuitively obvious fact that there is a natural
bijective correspondence between the $1$-extensions of a quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ and the extensions
of the pretrain $(F,\equiv^{F}_{1})$.

######  Lemma 11.2.

Let $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ be a quasitrain. If
the pretrain ${(H,\equiv^{H}_{1})}$ is an extension of $(F,\equiv^{F}_{1})$,
then there exists a unique quasitrain structure

$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})$

such that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
a subquasitrain of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$.

Notice that in this situation $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ is
a $1$-extension of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.

###### Proof of Lemma 11.2.

Dealing with the uniqueness first, we consider any such quasitrain
${\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})}$. Definition 10.1( )
‣ 10.1 determines the relation $\equiv^{H}_{0}$. For every $\mu\in[2,m]$ we
contend that

$\forall e^{\prime},e^{\prime\prime}\in
E(H)\,\,\,\bigl{[}e^{\prime}\equiv^{H}_{\mu}e^{\prime\prime}\,\,\,\Longleftrightarrow\,\,\,\exists
e_{\star},e_{\star\star}\in
E(F)\,\,\,\,e^{\prime}\equiv^{H}_{1}e_{\star}\equiv^{F}_{\mu}e_{\star\star}\equiv^{H}_{1}e^{\prime\prime}\bigr{]}\,.$
(11.1)

Indeed, let $\mu\in[2,m]$ and $e^{\prime},e^{\prime\prime}\in E(H)$. For the
backwards implication we just need to observe that because of Definition 10.1(
) ‣ 10.1 and Definition 10.6( ) ‣ 10.6 the formula
$e^{\prime}\equiv^{H}_{1}e_{\star}\equiv^{F}_{\mu}e_{\star\star}\equiv^{H}_{1}e^{\prime\prime}$
yields
$e^{\prime}\equiv^{H}_{\mu}e_{\star}\equiv^{H}_{\mu}e_{\star\star}\equiv^{H}_{\mu}e^{\prime\prime}$,
whence $e^{\prime}\equiv^{H}_{\mu}e^{\prime\prime}$.

For the forwards direction, we suppose
$e^{\prime}\equiv^{H}_{\mu}e^{\prime\prime}$ and take two edges
$e_{\star},e_{\star\star}\in E(F)$ which are in the same $1$-wagons of $H$ as
$e^{\prime}$, $e^{\prime\prime}$, respectively. Since $(H,\equiv^{H}_{1})$ is
an extension of $(F,\equiv^{F}_{1})$, Definition 6.2( ) ‣ 6.2 informs us that
such edges do indeed exist. Now
$e_{\star}\equiv^{H}_{1}e^{\prime}\equiv^{H}_{\mu}e^{\prime\prime}\equiv^{H}_{1}e_{\star\star}$
implies $e_{\star}\equiv^{H}_{\mu}e_{\star\star}$. Due to the fact that
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
a subquasitrain of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ we
can conclude $e_{\star}\equiv^{F}_{\mu}e_{\star\star}$, and altogether the
edges $e_{\star}$, $e_{\star\star}$ are as desired. This proves (11.1) and the
uniqueness of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
follows.

Addressing the claim on the existence of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ we
define $\equiv^{H}_{0}$ as demanded by Definition 10.1( ) ‣ 10.1 and for
$\mu\in[2,m]$ we define $\equiv^{H}_{\mu}$ by (11.1). We omit the easy proof
that these relations are indeed equivalence relations satisfying the clauses (
) ‣ 10.1 and ( ) ‣ 10.1 of Definition 10.1. ∎

The following special cases are going to be important soon.

######  Example 11.3.

Suppose that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ is an arbitrary
ordered quasitrain of height $m$ and that $(H,\equiv^{H}_{1})$ denotes the
ordered pretrain obtained from $(F,\equiv^{F}_{1})$ by wagon assimilation as
discussed in Example 6.4. Lemma 11.2 leads us to a (unique) ordered quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})$ of height $m$ which
is a $1$-extension of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.
It will be convenient to say that
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
arises from $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ by
assimilation of its $1$-wagons.

######  Example 11.4.

Consider an ordered hypergraph pair $(X,W)$ such that neither $X$ nor $W$ has
isolated vertices. Given an ordered quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ all of whose
$1$-wagons are order-isomorphic to $W$ we can form the ordered pretrain

$(G,\equiv^{G}_{1})=(F,\equiv^{F}_{1})\ltimes(X,W)$

as in Definition 6.6. Since $(G,\equiv^{G}_{1})$ is an extension of
$(F,\equiv^{F}_{1})$, Lemma 11.2 gives rise to a (unique) $1$-extension
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}}=(G,\equiv^{G}_{0},\ldots,\equiv^{G}_{m})$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.
We shall write $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}}=\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}\ltimes(X,W)$ for this construction (see Figure 11.1).

$X$$X$$X$$X$$1$$2$$3$$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
Figure 11.1. The train $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}\ltimes(X,W)$, where
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ is
$3$-partite and has the parameter $(\\{1,2\\},\\{3\\},\\{1\\})$. For space
reasons, the $3$-partite structure of $X$ is not shown in this figure.

######  Lemma 11.5.

If the quasitrain $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})$ of height $m$ is a
$1$-extension of the quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$, then for every
$\mu\in[m]$ the pretrain $(H,\equiv^{H}_{\mu})$ is an extension of
$(F,\equiv^{F}_{\mu})$.

###### Proof.

Given $\mu\in[m]$ we check the three conditions in Definition 6.2. There is no
problem with ( ) ‣ 6.2. For ( ) ‣ 6.2 we consider an arbitrary $\mu$-wagon
$W_{\mu}$ of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
and take an edge $e\in E(W_{\mu})$. Since the $1$-wagon of $e$ contracts to
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$,
there is an edge $e^{\prime}\in E(F)$ with $e^{\prime}\equiv^{H}_{1}e$. In
particular, we have $e^{\prime}\in E(W_{\mu})\cap E(F)$, meaning that
$W_{\mu}$ indeed contracts.

Proceeding with ( ) ‣ 6.2 we let $\overline{W}^{\prime}_{\mu}$,
$\overline{W}^{\prime\prime}_{\mu}$ be the contractions of two distinct
$\mu$-wagons $W^{\prime}_{\mu}$, $W^{\prime\prime}_{\mu}$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$.
Now $V(\overline{W}^{\prime}_{\mu})\cap
V(\overline{W}^{\prime\prime}_{\mu})\subseteq V(W^{\prime}_{\mu})\cap
V(W^{\prime\prime}_{\mu})$ is clear and for the reverse inclusion we look at
an arbitrary vertex $x\in V(W^{\prime}_{\mu})\cap V(W^{\prime\prime}_{\mu})$.
Pick two edges $e^{\prime}\in E(W^{\prime}_{\mu})$, $e^{\prime\prime}\in
E(W^{\prime\prime}_{\mu})$ with $x\in e^{\prime}\cap e^{\prime\prime}$ and
denote the $1$-wagons of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
these edges belong to by $W^{\prime}_{1}$, $W^{\prime\prime}_{1}$,
respectively. Since $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ is
a $1$-extension of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$,
the contractions $\overline{W}^{\prime}_{1}$,
$\overline{W}^{\prime\prime}_{1}$ of these wagons satisfy

$x\in V(W^{\prime}_{1})\cap
V(W^{\prime\prime}_{1})=V(\overline{W}^{\prime}_{1})\cap
V(\overline{W}^{\prime\prime}_{1})$

and, consequently, there exist edges $e_{\star}\in
E(\overline{W}^{\prime}_{1})$, $e_{\star\star}\in
E(\overline{W}^{\prime\prime}_{1})$ with $x\in e_{\star}\cap e_{\star\star}$.
As $e^{\prime}$ and $e_{\star}$ are in the same $1$-wagon of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$,
they are also in the same $\mu$-wagon, for which reason $e_{\star}\in
E(W^{\prime}_{\mu})\cap E(F)=E(\overline{W}^{\prime}_{\mu})$. Similarly we
have $e_{\star\star}\in E(\overline{W}^{\prime\prime}_{\mu})$ and altogether
$x\in V(\overline{W}^{\prime}_{\mu})\cap V(\overline{W}^{\prime\prime}_{\mu})$
follows. ∎

In general, $1$-extensions of trains are only known to be quasitrains, but
they may fail to be trains. This is for the reason that two edges of an
enlarged $1$-wagon might intersect each other in a ’wrong’ vertex class. The
lemma that follows shows that, actually, this is the only obstacle.

######  Lemma 11.6.

Suppose that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ is an $f$-partite
train of height $m$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
A\hfil$\crcr}}}=(A_{1},\ldots,A_{m})$ and that the $f$-partite quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})$ is a $1$-extension
of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.
If every $1$-wagon $X$ of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
has the property that $e^{\prime}\cap e^{\prime\prime}\subseteq V_{A_{1}}(H)$
holds for any two distinct edges $e^{\prime},e^{\prime\prime}\in E(X)$, then
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ is
a train with parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$.

###### Proof.

We need to check that Definition 10.4( ) ‣ 10.4 holds for
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$.
For $\mu=1$ this was stated as a hypothesis, so suppose $\mu\in[2,m]$ from now
on. Let $W^{\prime}_{\mu-1}$ and $W^{\prime\prime}_{\mu-1}$ be two distinct
$(\mu-1)$-wagons of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
contained in the same $\mu$-wagon and denote their contractions to
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ by
$\overline{W}^{\prime}_{\mu-1}$, $\overline{W}^{\prime\prime}_{\mu-1}$. Lemma
11.5 informs us that $(H,\equiv^{H}_{\mu-1})$ is an extension of
$(F,\equiv^{F}_{\mu-1})$ and thus we have indeed

$V(W^{\prime}_{\mu-1})\cap
V(W^{\prime\prime}_{\mu-1})=V(\overline{W}^{\prime}_{\mu-1})\cap
V(\overline{W}^{\prime\prime}_{\mu-1})\subseteq V_{A_{\mu}}(F)\subseteq
V_{A_{\mu}}(H)\,.\qed$

Taking a $1$-extension can affect the $\mathfrak{girth}$ of a quasitrain, for
the new $1$-wagons might contain shorter cycles than the original $1$-wagons.
The next result shows, that if there are no problems with $1$-wagons, then
$1$-extensions inherit the $\mathfrak{girth}$ of the original quasitrain.

######  Lemma 11.7.

Suppose that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}=(g_{1},\dots,g_{m})\in\mathfrak{M}$ is a sequence of some
positive length $m$, that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}}=(F,\equiv^{F}_{0},\ldots,\equiv^{F}_{m})$ is a quasitrain with
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$
and that $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}}=(H,\equiv^{H}_{0},\ldots,\equiv^{H}_{m})$ is a $1$-extension
of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$.
If every $1$-wagon $X$ of $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
satisfies $\mathrm{girth}(X)>g_{1}$, then
$\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$.

###### Proof.

Given $\mu\in[m]$ and a $\mu$-wagon $W_{\mu}$ of
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ we
are to prove that
$\mathfrak{girth}(W_{\mu},\equiv^{W_{\mu}}_{\mu-1})>g_{\mu}$. In the special
case $\mu=1$ this was stated as an assumption, so it remains to consider the
case $\mu\in[2,m]$. Assume for the sake of contradiction that for some
$n\in[2,g_{\mu}]$ there exists an $n$-cycle of distinct $(\mu-1)$-wagons

$\mathscr{C}=W^{1}_{\mu-1}q_{1}\ldots W^{n}_{\mu-1}q_{n}$

all of which are contained in $W_{\mu}$. Owing to Lemma 11.5 these
$(\mu-1)$-wagons contract to $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
and the connecting vertices $q_{1},\ldots,q_{n}$ are in $V(F)$. Therefore, the
contraction $\overline{W}_{\mu}$ of $W_{\mu}$ fails to have the property
$\mathfrak{girth}(\overline{W}_{\mu},\equiv^{\overline{W}_{\mu}}_{\mu-1})>g_{\mu}$,
contrary to $\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$. ∎

Let us state an immediate consequence of Lemma 11.6 and Lemma 11.7.

######  Corollary 11.8.

If $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
g\hfil$\crcr}}}\in\mathfrak{M}$ is a nonempty sequence,
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$
denotes a train of height $|\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}|$
with parameter $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
and $\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
F\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$,
and the quasitrain $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$
arises from $\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle F\hfil$\crcr}}}$ by
assimilation of its $1$-wagons, then
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle H\hfil$\crcr}}}$ is
again a train of height $m$ with parameter
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle A\hfil$\crcr}}}$
and $\mathfrak{girth}(\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
H\hfil$\crcr}}})>\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle g\hfil$\crcr}}}$. ∎

We conclude this subsection with a brief discussion of disjoint unions of
quasitrains. Suppose first that
$\\{\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}}_{j}\colon j\in J\\}$ is a family of mutually vertex-disjoint
$f$-partite quasitrains of the same height $m\in{\mathds{N}}$, say
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}}_{j}=(G_{j},\equiv^{j}_{0},\ldots,\equiv^{j}_{m})$ for every
$j\in J$. By the union of this family we mean the $f$-partite quasitrain
$\ThisStyle{\vbox{\halign{#\cr$\SavedStyle\mkern
0.8mu\cleaders\hbox{$\smash{\scalebox{0.55}{$\SavedStyle\relbar$}}\mkern-4.0mu$}\hfill\raisebox{0.0pt}[2.0pt][1.0pt]{\scalebox{0.55}{$\SavedStyle\rightharpoonup$}}\mkern
0.8mu$\crcr\kern-1.0pt\nointerlineskip\cr$\hfil\SavedStyle
G\hfil$\crcr}}}=(G,\equiv^{G}_{0},\ldots,\equiv^{G}_{m})$ with

$V(G)=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{