# No universal group in a cardinal

Saharon Shelah  Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 91904, Israel
and
Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA<EMAIL_ADDRESS>http://shelah.logic.at

(Date: April 21, 2014)

###### Abstract.

For many classes of models there are universal members in any cardinal
$\lambda$ which “essentially satisfies GCH, i.e. $\lambda=2^{<\lambda}$”. But
if the class is “complicated enough”, e.g. the class of linear orders, we know
that if $\lambda$ is “regular and not so close to satisfying GCH” then there
is no universal member. Here we find new sufficient conditions (which we call
the olive property), not covered by earlier cases (i.e. fail the so-called
${\rm SOP}_{4}$). The advantage of those conditions is witnessed by proving
that the class of groups satisfies one of those conditions.

###### Key words and phrases:

model theory, universal models, the olive property, group theory, non-
structure, classification theory

###### 2010 Mathematics Subject Classification:

Primary: 03C55, 20A15; Secondary: 03C45, 03E04

The author thanks the Israel Science Foundation for partial support of this
work. Grant No. 1053/11. The author thanks Alice Leonhardt for the beautiful
typing. First typed March 27, 2013. Publication 1029.

Anotated Content

§0 Introduction, (labels y,z), pg. ‣ 0\. Introduction

§1 The olive property, (label d), pg.1

  1. [We give definitions of some versions of the olive property and give an example failing the ${\rm SOP}_{4}$. We phrase relevant set theoretic conditions like ${\rm Qr}_{1}$ (slightly weaker than those used earlier). Then we give complete proof using ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ to deduce ${\rm Univ}(\chi_{1},\lambda,\mathbb{k})\geq\chi_{2}$ so no universal in the class $\mathbb{k}$ in the cardinal $\lambda$, when $\mathbb{k}$ has the olive property.]

§2 The class of groups have the olive property, (label s), pg.2

  1. [We prove the stated result. We also deal with the non-existence of universal structures for pairs of classes, e.g. the pair (locally finite groups, groups).]

§3 Concluding Remarks, (label m), pg. 3

  1. [We consider some generalizations of the properties, but no clear gain.]

##  0\. Introduction

###  0(A). Background and open questions

On history see Kojman-Shelah [KjSh:409] and later Dzamonja [Mir05]. Recall
that if $\lambda=2^{<\lambda}>\aleph_{0}$ then many classes have a universal
in $\lambda$, so assuming GCH, we know when there is a universal model in
every $\lambda>\aleph_{0}$.

For transparency we consider a first order countable $T$. Recall that on the
one hand Kojman-Shelah [KjSh:409] show that if $T$ is the theory of dense
linear orders or just $T$ has the strict order property, then $T$ fails (in a
strong way) to have a universal in regular cardinals in which cardinal
arithmetic is “not close to GCH”; (for regular $\lambda$ this means there is a
regular $\mu$ such that $\mu^{+}<\lambda<2^{\mu}$, for singular $\lambda$ we
need of course $\lambda<2^{<\lambda}$ and a very weak pcf condition).

By [Sh:500], we can weaken “the strict order property” to the 4-strong order
property ${\rm SOP}_{4}$.

Natural questions are (we shall address some of them):

######  Question 0.1.

1) Is there a weaker condition (on $T$) than ${\rm SOP}_{4}$ which suffice?

2) Can we find a best one?

3) Can we find such a condition satisfied for some theory $T$ which is ${\rm
NSOP}_{3}$?

######  Question 0.2.

1) Is there $T$ with the class ${\rm Univ}(T)\backslash(2^{\aleph_{0}})^{+}$
strictly smaller than the one for linear order, see 0.12(3); we better
restrict ourselves to regular cardinals above $2^{\aleph_{0}}$?

2) Can we get the above to be $\\{\lambda:\lambda=2^{<\lambda}\\}$?

3) What about singular cardinals?

######  Question 0.3.

1) Is it consistent that the class of linear order has a universal member in
$\lambda$ such that $2^{<\lambda}>\lambda>2^{\aleph_{0}}$ (for
$\lambda=\aleph_{1}<2^{\aleph_{0}}$, yes, [Sh:100]).

2) Similarly for some theory with ${\rm SOP}_{4}$ or the olive property.

Recall that by Shelah-Usvyatsov [ShUs:789] the class of groups has ${\rm
NSOP}_{4}$ but has ${\rm SOP}_{3}$, so it was not clear where it stands.

######  Question 0.4.

1) Where does the class of group stand (concerning the existence of a
universal member in a cardinal)?

2) Is it consistent that there is a universal locally finite group of
cardinality $\aleph_{1}$? of cardinality $\beth_{\omega}$? of other
cardinality $\lambda<\lambda^{\aleph_{0}}$?

Recall (Grossberg-Shelah [GrSh:174]) if $\mu$ is strong limit of cofinality
$\aleph_{0}$ above a a compact cardinal, then there is a universal locally
finite group of cardinality $\mu$ but if $\mu=\mu^{\aleph_{0}}$ then there is
no one.

Concerning singulars

######  Question 0.5.

Does $\theta={\rm cf}(\theta)$ and $\theta^{+2}<{\rm
cf}(\lambda)<\lambda<2^{\theta}$ implies $\lambda<{\rm univ}(\lambda,T)$?

######  Question 0.6.

0) Characterize the failure of the criterion of [Sh:457], Džamonja-Shelah
[DjSh:614](for consistency).

1) Does ${\rm SOP}_{3}$ (or something weaker) suffice for no universal in
$\lambda$ when $\mu=\mu^{<\mu}\ll\lambda<2^{\mu}$?

2) Which theories $T$ fails to have a universal in $\lambda$ when
$\lambda=\mu^{++}=2^{\mu}<2^{\mu^{+}}$.

3) Weaker properties of $T$ for no universal in
$\lambda,\mu=\mu^{<\mu}\ll\lambda<2^{\mu}$.

4) Sort out the variants of the olive property.

######  Discussion 0.7.

The case $\lambda=\mu^{+},\lambda<2^{\mu}$ and $2^{<\mu}\leq\lambda$ (e.g. for
transparency $\mu=\mu^{<\mu}$) is not resolved as we do not necessarily have
$\bar{C}=\langle C_{\delta}:\delta\in S^{\lambda}_{\mu}\rangle$ guessing
clubs.

Earlier if $\mu=2^{\kappa}$, so $\mu$ not strong limit, in the case of failure
there was a sequence $\langle\Lambda_{\delta}:\delta\in
S^{\lambda}_{\mu}\rangle,\Lambda_{\delta}\subseteq{}^{(C_{\delta})}\mu$ of
cardinality $\lambda$ such that for every sequence
$\langle\eta_{\delta}\subseteq{}^{(C_{\delta})}\mu:\delta\in
S^{\lambda}_{\mu}\rangle$ for some club $E$ of $\lambda$ for every $\delta\in
E\cap S^{\lambda}_{\mu}$ for some $\nu\in\Lambda_{\delta}$ the functions
$\eta_{\delta},\nu$ agree on $E\cap{\rm nacc}(C_{\delta})$.

Using more complicated $T$ we can replace ${}^{C_{\delta}}\mu$ by
${}^{(C_{\delta}\times D_{\delta})}\mu$ so the agreement above is on
$(E\cap{\rm nacc}(C_{\delta}))\times(E\cap{\rm nacc}(C_{\delta}))$ but of
unclear value.

See lately [Sh:F1330], more on consistency (after the present work) see
[Sh:F1414] on 0.2, and more on ZFC results in [Sh:F1425].

###  0(B). What is accomplished

What do we achieve? We introduce the “olive property” which suffice for the
class to have a universal member in $\lambda$ only if $\lambda$ is “close to
satisfying G.C.H.”, similarly to the linear order case. This condition is
weaker than ${\rm SOP}_{4}$, hence gives a positive answer to 0.8(1). But the
condition implies ${\rm SOP}_{3}$ so it does not answer 0.8(3), also it is
totally unclear whether it is best in any sense and whether its negation has
interesting consequences.

However, it answers 0.4(1) to a large extent because the class of groups has
the olive property and we can also deal with locally finite groups; see §2.
Also we try to formalize conditions sufficient for non-existence, see 1.6 and
see more in §3. As the reader may find the definition of the (variants of the)
olive property opaque, we define a simple case used for the class of groups,
and the reader then may look first at the class of groups in §2.

######  Definition 0.8.

A (first order) universal theory $T$ has the olive property when there are
$(\varphi_{0},\varphi_{1},\psi)$ and model ${\mathfrak{C}}$ of $T$ such that:

  1. $(a)$

for some
$m,\varphi_{0}=\varphi_{0}(\bar{x}_{[m]},\bar{y}_{[m]}),\varphi_{1}=\varphi_{1}(\bar{x}_{[m]},\bar{y}_{[m]}),\psi=\psi(\bar{x}_{[m]},\bar{y}_{[m]},\bar{z}_{[m]})$
are quantifier free formulas (and $\bar{x}_{[m]},\bar{y}_{[m]},\bar{z}_{[m]}$
are $m$-tuples of variables, see 0.10 below)

  2. $(b)$

for every $k$ and $\bar{f}=\langle f_{\alpha}:\alpha<k\rangle,f_{\alpha}$ is a
function from $\alpha$ to $\\{0,1\\}$ we can find
$\bar{a}_{\alpha}\in{}^{m}{{\mathfrak{C}}}$ for $\alpha<k$ such that:

    1. $(\alpha)$

$\varphi_{\iota}[\bar{a}_{\alpha},\bar{a}_{\beta}]$ for $\alpha<\beta<k$ when
$\iota=f_{\beta}(\alpha)$

    2. $(\beta)$

$\psi[\bar{a}_{\alpha},\bar{a}_{\beta},\bar{a}_{\gamma}]$ when
$\alpha<\beta<\lambda$ and $f_{\gamma}{\restriction}[\alpha,\beta]$ is
constantly 0

  3. $(c)$

there are no $\bar{a}_{\ell}\in{}^{m}{\mathfrak{C}}$ for $\ell=0,1,2,3$ such
that the following are111in the class of groups, in clause
$(\alpha),\varphi_{0}[\bar{a}_{0},\bar{a}_{1}),\varphi_{1}[a_{1},a_{2}],\varphi_{1}[a_{1},\bar{a}_{3}]$
suffice satisfied in ${\mathfrak{C}}$

    1. $(\alpha)$

$\varphi_{0}[\bar{a}_{0},\bar{a}_{\ell}]$ for
$\ell=1,2,3,\varphi_{1}[\bar{a}_{1},\bar{a}_{\ell}]$ for $\ell=1,2$ and
$\varphi_{0}[\bar{a}_{2},\bar{a}_{3}]$

    2. $(\beta)$

$\psi[\bar{a}_{0},\bar{a}_{2},\bar{a}_{3}]$

######  Concluding Remarks 0.9.

Concerning some things not addressed here.

1) Concerning the proof here of “there is no universal” we can carry it via
defining invariants parallel to Kojman-Shelah [KjSh:409] such that (for
transparency $\lambda$ is regular uncountable, see 0.11(5),(7))

  1. $(*)$

$(a)\quad$ if $M\in{\rm Mod}_{T,\lambda}$ then ${\rm INV}_{\lambda}(M)$ is a
set of cardinality $\leq\lambda$ or just

$\leq\chi<2^{\lambda}$

  2. $(b)\quad$ if $M_{1},M_{2}\in{\rm Mod}_{T,\lambda}$ and $M_{1}$ is embeddable into $M_{2}$ then

${\rm INV}_{\lambda}(M_{1})\subseteq{\rm INV}_{\lambda}(M_{2})$

  3. $(c)\quad$ there is a set of $2^{\lambda}$ objects $\mathbb{x}$ such that $(\exists M\in{\rm Mod}_{T,\lambda})(\mathbb{x}\in{\rm INV}_{\lambda}(M))$.

2) We can use more complicated versions of the olive property. In the proof we
use one $\delta$ and then one $\alpha\in{\rm nacc}(C_{\delta})\cap E$ (or
less), but we may use several $\alpha$’s getting more complicated versions.
This will become more pressing if we have a complimentary property,
guaranteeing “no universal” or some variant.

###  0(C). Preliminaries

######  Notation 0.10.

1) Let $\bar{x}_{[I]}=\langle x_{t}:t\in I\rangle$ and similarly
$\bar{y}_{[I]},\bar{x}_{[I],\alpha}$, etc. and $\bar{x}_{[I],\ell}=\langle
x_{t,\ell}:t\in I\rangle$.

2) For a first order complete $T,{\mathfrak{C}}_{T}$ is the “monster model of
$T$”.

######  Definition 0.11.

1) For a set $A,|A|$ is its cardinality but for a structure $M$ its
cardinality is $\|M\|$ while its universe is $|M|$; this apply e.g. to groups.

2) We use $G,H$ for groups, $M,N$ for general models.

3) Let ${\mathfrak{k}}$ denote a pair
$(K_{{\mathfrak{k}}},\leq_{{\mathfrak{k}}})$, may say a class
${\mathfrak{k}}$, where:

  1. $(a)$

$K_{{\mathfrak{k}}}$ is a class of $\tau_{{\mathfrak{k}}}$-structures where
$\tau_{{\mathfrak{k}}}$ is a vocabulary

  2. $(b)$

$\leq_{{\mathfrak{k}}}$ is a partial order on $K_{{\mathfrak{k}}}$ such that
$M\leq_{{\mathfrak{k}}}N\Rightarrow M\subseteq N$

  3. $(c)$

both $K_{{\mathfrak{k}}}$ and $\leq_{{\mathfrak{k}}}$ are closed under
isomorphisms.

4) We say $f:M\rightarrow N$ is a $\leq_{{\mathfrak{k}}}$-embedding when $f$
is an isomorphism from $M$ onto some $M_{1}\leq_{{\mathfrak{k}}}N$.

5) If $T$ is a first order theory then ${\rm Mod}_{T}$ is the pair
$(\text{Mod}_{T},\leq_{T})$ where $\text{ mod}_{T}$ is the class of models of
$T$ and $\leq_{T}$ is: $\prec$ if $T$ is complete, $\subseteq$ if $T$ is not
complete.

6) We may write $T$ instead of ${\rm Mod}_{T}$, e.g. in Definition 0.12 below.

7) For a class $K$ of structures $K_{\lambda}=\\{M\in K:\|M\|=\lambda\\}$.

######  Definition 0.12.

1) For a class ${\mathfrak{k}}$ and a cardinal $\lambda$, a set
$\\{M_{i}:i<i^{*}\\}$ of models from $K_{{\mathfrak{k}}}$, is jointly
$(\lambda,{\mathfrak{k}})$-universal when for every $N\in K_{{\mathfrak{k}}}$
of size $\lambda$, there is an $i<i^{*}$ and an
$\leq_{{\mathfrak{k}}}$-embedding of $N$ into $M_{i}$.

2) For ${\mathfrak{k}}$ and $\lambda$ as above, let (if $\mu=\lambda$ we may
omit $\mu$)

$\begin{array}[]{clcr}{\rm
univ}(\lambda,\mu,{\mathfrak{k}}):=&\min\\{|{\mathscr{M}}|:{\mathscr{M}}\text{
is a family of members of }K_{{\mathfrak{k}}}\text{ each}\\\ &\text{ of
cardinality }\leq\mu\text{ which is jointly}\\\
&{\mathfrak{k}}\text{-universal for }\lambda\\}\end{array}$

Let ${\rm Univ}({\mathfrak{k}})=\\{\lambda:{\rm
univ}(\lambda,{\mathfrak{k}})=1\\}$.

3) For a pair $\bar{{\mathfrak{k}}}=({\mathfrak{k}}_{1},{\mathfrak{k}}_{2})$
of classes with
${\mathfrak{k}}_{\iota}=(K_{{\mathfrak{k}}_{\iota}},\leq_{{\mathfrak{k}}_{\iota}})$
as in 0.11(3) for $\iota=1,2$ such that $K_{{\mathfrak{k}}_{1}}\subseteq
K_{{\mathfrak{k}}_{2}}$, let ${\rm univ}(\lambda,\mu,\bar{{\mathfrak{k}}})$ be
the minimal $|{\mathscr{M}}|$ such that ${\mathscr{M}}$ is a family of members
of $K_{{\mathfrak{k}}_{2}}$ each of cardinality $\mu$ such that every $M\in
K_{{\mathfrak{k}}_{1}}$ of cardinality $\lambda$ can be
$\leq_{{\mathfrak{k}}_{2}}$-embedded into some member of ${\mathscr{M}}$.

Dealing with a.e.c.’s (see [Sh:h])

######  Definition 0.13.

1) We say that a formula $\varphi=\varphi(\bar{x}_{[I]})$, in any logic, is
${\mathfrak{k}}$-upward preserved when
$\tau_{\varphi}\subseteq\tau_{{\mathfrak{k}}}$ and if
$M\leq_{{\mathfrak{k}}}N$ and $\bar{a}\in{}^{I}M$ then
$M\models\varphi[\bar{a}]$ implies $N\models\varphi[\bar{a}]$.

2) For $\bar{{\mathfrak{k}}}$ as in 0.12(3) we say a pair
$\bar{\varphi}(\bar{x}_{[I]})=(\varphi_{1}(\bar{x}_{[I]}),\varphi_{2}(\bar{x}_{[I]}))$
is $\bar{{\mathfrak{k}}}$-upward preserving when
$\tau_{\varphi_{1}}\cup\tau_{\varphi_{2}}\subseteq\tau_{{\mathfrak{k}}_{\iota}}$
and if $M_{\iota}\in K_{{\mathfrak{k}}_{\iota}}$ for
$\iota=1,2,\bar{a}\in{}^{I}(M_{1})$ and $M_{1}\leq_{{\mathfrak{k}}_{2}}M_{2}$
then $M_{1}\models\varphi_{1}[\bar{a}]$ implies
$M_{2}\models\varphi_{1}[\bar{a}]$.

3) In part (2), if $\varphi_{0}=\varphi_{1}$ then we may write $\varphi$
instead of $\bar{\varphi}$. Saying a sequence $\bar{\psi}$ is
${\mathfrak{k}}$-upward preserving means every formula appearing in
$\bar{\psi}$ is ${\mathfrak{k}}$-upward preserving.

######  Definition 0.14.

1) For an ideal $J$ on a set $A$ and a set $B$ let $\mathbb{U}_{J}(B)={\rm
Min}\\{|{{\mathscr{P}}}|:{{\mathscr{P}}}$ is a family of subsets of $B$, each
of cardinality $\leq|A|$ such that for every function $f$ from $A$ to $B$ for
some $u\in{{\mathscr{P}}}$ we have $\\{a\in A:f(a)\in u\\}\in J^{+}\\}$.

2) For an ideal $J$ on a set $A$, cardinal $\theta$ and set $B$ let
$\mathbb{U}^{\theta}_{J}(B)={\rm
Min}\\{|{\mathscr{P}}|:{\mathscr{P}}\subseteq[B]^{\leq|A|}$ and if
$f\in{}^{A}({}^{\theta}B)$ then for some $u\in{\mathscr{P}}$ we have $\\{a\in
A:{\rm Rang}(f(a))\subseteq u\\}\in J^{+}\\}$. So
$\mathbb{U}^{\theta}_{J}(B)\leq\mathbb{U}_{J}(|N|^{\theta})$.

3) Clearly only $|B|$ matters so we normally write $\mathbb{U}_{J}(\lambda)$,
(see on it [Sh:589]).

##  1\. The Olive property

######  Definition 1.1.

1) (Convention)

  1. $(a)$

Let $T$ be a first order theory and ${\mathfrak{C}}={\mathfrak{C}}_{T}$ a
monster for $T$

  2. $(b)$

$(\alpha)\quad\Delta\subseteq{\mathbb{L}}(\tau_{T})$ a set of formulas

  3. $(\beta)\quad$ omitting $\Delta$ means $\Delta={\mathbb{L}}(\tau_{T})$ if $T$ is complete, $\Delta$ = set of quantifiers

free formula otherwise, and we may write ${\rm qf}$ instead of $\Delta$

  4. $(c)$

$(\alpha)\quad m$ and $n\geq k_{\iota}\geq 2$ for $\iota=0,1,n\geq
k_{0}+k_{1}\geq 3,\eta\in{}^{n}2$ are such that

$\eta(0)=0$ and $\eta^{-1}\\{0\\}$ is not an initial segment

  5. $(\beta)\quad$ if $\eta(\ell)=\ell\mod 2$ for $\ell<k$ we may write $n$ instead of $\eta$

  6. $(d)$

$(\alpha)\quad$ if $\bar{k}=(k_{0},k_{1}),k_{1}\leq k_{0}+1\leq k_{1}+1$ we
may write

$k_{0}+k_{1}$ instead of $\bar{k}$ and let $k(\iota)=k_{\iota}$ for
$\iota=0,1$.

  7. $(\beta)\quad$ omitting $m$ means some $m$

  8. $(\gamma)\quad$ omitting $n,\eta,\bar{k}$ means $n=3,\eta=\langle 0,1,0\rangle,\bar{k}=(2,1)$) so for some $m$

  9. $(e)$

$(\alpha)\quad$ below we may write $\psi_{\iota}=\psi_{\iota,k_{\iota}}$ and
$\varphi_{\iota}=\psi_{\iota,1}$ for $\iota=0,1$

  10. $(\beta)\quad$ if $\varphi_{0}=\varphi_{1}=\varphi$ we may write $\varphi$;

  11. $(\gamma)\quad$ we may omit $\psi_{3,k}$ when it is a logically true formula.

2) We say $T$ has the $(\Delta,\eta,\bar{k},m)$-olive property when there is a
pair $(\bar{\psi}_{0},\bar{\psi}_{1})$ of sequences of formulas from $\Delta$
witnessing it, see (3).

3) We say $(\bar{\psi}_{0},\bar{\psi}_{1})$ witness the
$(\Delta,\eta,\bar{k},m)$-olive property (for $T$, with the convention above)
when :

  1. $(a)$

$\bar{\psi}_{\iota}=\langle\psi_{\iota,k}(\bar{x}_{[m],0},\dotsc,\bar{x}_{[m],k}):k=1,\dotsc,k_{\iota}\rangle$
for $\iota=0,1$ with $\psi_{\iota,k}\in\Delta$

  2. $(b)_{\lambda}$

for every $\bar{f}=\langle f_{\alpha}:\alpha<\lambda\rangle$ with $f_{\alpha}$
a function from $\alpha$ to $\\{0,1\\}$, we can find
$\bar{a}_{\alpha}\in{}^{m}{{\mathfrak{C}}}$ for $\alpha<\lambda$
such222Actually clause $(\alpha)$ is a specific case of clause $(\beta)$
provided that in clause $(\beta)$ we allow $k=1$. Similarly for clauses
$(c)(\alpha),(\beta)$. that:

    1. $(\alpha)$

$\varphi_{\iota}[\bar{a}_{\alpha},\bar{a}_{\beta}]$ for $\alpha<\beta<\lambda$
when $\iota=f_{\beta}(\alpha)$, see 1.1$(1)(e)(\beta)$

    2. $(\beta)$

$\psi_{\iota,k}(\bar{a}_{\alpha_{0}},\dotsc,\bar{a}_{\alpha_{k-1}},\bar{a}_{\beta})$
when $k\in\\{2,\dotsc,k_{\iota}-1\\}$ and
$\alpha_{0}<\ldots<\alpha_{k-1}<\beta<\lambda$ and
$f_{\beta}{\restriction}[\alpha_{0},\alpha_{k-1}]$ is constantly $\iota$, so
when $k=1$, it holds trivially

  3. $(c)$

there are no $\bar{a}_{\ell}\in{}^{m}{{\mathfrak{C}}}$ for $\ell<n+1$ such
that:

    1. $(\alpha)$

$\varphi_{\iota}[\bar{a}_{i},\bar{a}_{j}]$ for $i<j<n+1$ and $\eta(i)=\iota$

    2. $(\beta)$

if $\iota\in\\{0,1\\},k\in\\{2,\dotsc,k_{\iota}\\}$ and
$\ell_{0}<\ldots<\ell_{k-1}$ are from $\\{\ell<n:\eta(\ell)=\iota\\}$ and
$\ell_{k-1}<\ell\leq n$ then
$\psi_{\iota,k}[\bar{a}_{\ell_{0}},\dotsc,\bar{a}_{\ell_{k-1}},\bar{a}_{\ell}]$.

######  Remark 1.2.

This fits the classification of properties of such $T$ in [Sh:702, 5.15-5.23].

######  Definition 1.3.

1) Let $K$ be a universal class of $\tau$-models. We say $K$ has the
$\lambda-(\eta,\bar{k},m)$-olive property when that some quantifier free
$(\bar{\psi}_{0},\bar{\psi}_{1})$ witnessing it, that is,
$(a)+(b)_{\lambda}+(c)$ holds (replacing ${\mathfrak{C}}_{T}$ by “in some
$M\in K$”).

2) We say that an a.e.c.
${\mathfrak{k}}=(K_{{\mathfrak{k}}},\leq_{{\mathfrak{k}}})$ has the
$\lambda-(\eta,\bar{k},m)$-property when : there are
$\bar{\psi}_{0},\bar{\psi}_{1}$ which are ${\mathfrak{k}}$-upward preserved
formulas in any logic (see 0.13) and $(a)+(b)_{\lambda}+(c)$ of 1.1 holds,
replacing ${\mathfrak{C}}={\mathfrak{C}}_{T}$ by “some ${\mathfrak{C}}\in
K_{{\mathfrak{k}}}$ of cardinality $\lambda$”.

######  Remark 1.4.

1) Note that for $T$ first order complete, ${\mathfrak{k}}={\rm
Mod}_{T}=(\text{mod}_{T},\prec)$, Definition 1.3(2) gives Definition 1.1 and
for $T$ first order universal not complete, ${\mathfrak{k}}={\rm
Mod}_{T}=(\text{mod}_{T},\subseteq)$, Definition 1.3(2) gives Definition 1.1.
Similarly for Definition 1.3(1).

2) Of course, for $T$ first order, the $\lambda$ does not matter.

######  Claim 1.5.

Assume $n\geq k_{0}+k_{1}\geq 3,\eta\in{}^{n}2$ and
$|\eta^{-1}\\{\iota\\}|\geq k_{\iota}\geq 1$ for $\iota=0,1$ then there is a
complete first order countable $T$ having the $(\eta,\bar{k},1)$-olive
property but $T$ is ${\rm NSOP}_{4}$ and is categorical in $\aleph_{0}$.

###### Proof..

Let $\tau=\\{P,Q_{0},Q_{1}\\}$ where $P$ is a binary predicate and $Q_{\iota}$
is a $(k_{\iota}+1)$-place predicates. Let $T^{0}_{\eta,\bar{k}}$ be the
following universal theory in ${\mathbb{L}}(\tau)$:

  1. $(*)_{1}$

a $\tau$-model $M$ is a model of $T^{0}_{\eta,\bar{k}}$ iff we cannot embed
$N^{*}_{\eta,\bar{k}}$ into $M$ where

  2. $\oplus\quad N^{*}_{\eta,\bar{k}}$ is the $\tau$-model with universe $\\{a_{0},\dotsc,a_{n}\\}$ as in $(c)(\alpha),(\beta)$ from

Definition 1.1(3) for
$\varphi(x_{0},x_{1})=P(x_{0},x_{1}),\psi_{\iota}(x_{0},\dotsc,x_{k(\iota)})=$

$Q_{\iota}(x_{0},\dotsc,x_{k(\iota)})$ recalling 1.1$(1)(e)(\gamma)$.

Now

  1. $(*)_{2}$

$T^{0}_{\eta,\bar{k}}$ has the JEP and amalgamation property by disjoint
union.

[Why? Assume $M_{0}\subseteq M_{1},M_{0}\subseteq M_{2}$ are models of $T_{0}$
(but abusing notation we allow $M_{0}$ to be empty) and
$|M_{1}|\cap|M_{2}|=|M_{0}|$, we define $M=M_{1}\cup M_{2}$ that is

  1. $(*)_{2.1}$

$(a)\quad|M|=|M_{1}|\cup|M_{2}|$

  2. $(b)\quad P^{M}=P^{M_{1}}\cup P^{M_{2}}$

  3. $(c)\quad Q^{M}_{\iota}=Q^{M_{1}}_{\iota}\cup Q^{M_{2}}_{\iota}$ for $\iota=1,2$.

So $M$ is a $\tau$-model, it is a model of $T$ as in (b) any pair of elements
belongs to a relation.]

  1. $(*)_{3}$

$T_{\eta,\bar{k}}$, the model completion of $T^{0}_{\eta,\bar{k}}$, is well
defined and has elimination of quantifiers.

[Why? As $\tau$ is finite with no function symbols and $(*)_{2}$.]

  1. $(*)_{4}$

$T_{\eta,\bar{k}}$ is ${\rm NSOP}_{4}$ (see [Sh:500, 2.5]).

[Why? Because

  1. $(*)_{4.1}$

if (A) then (B) where:

    1. $(A)$

$(a)\quad A_{0},A_{1},A_{2},A_{3}$ are disjoint sets

    2. $(b)\quad M_{\ell}$ is a model of $T^{0}_{\eta,\bar{k}}$ with universe $A_{\ell}$ for $\ell=0,1,2,3$

    3. $(c)\quad$ if $\\{\ell(1),\ell(2)\\}\in{\mathscr{W}}:=\\{\\{0,1\\},\\{1,2\\},\\{2,3\\},\\{3,4\\}\\}$ then

$M_{\\{\ell(1),\ell(2)\\}}$ is a model of $T^{0}_{k,n}$ with universe
$A_{\ell(1)}\cup A_{\ell(2)}$

extending $M_{\ell(1)}$ and $M_{\ell(2)}$

    4. $(B)$

$M=\cup\\{M_{\\{\ell(1),\ell(2)\\}}:\\{\ell(1),\ell(2)\\}\in{\mathscr{W}}\\}$
where the union is defined as in the proof of $(*)_{2}$, is a model of
$T^{0}_{k,n}$ extending all of them.]

[Why? Clearly $M$ is a $\tau$-model and if $f$ embeds $N^{*}_{\eta,\bar{k}}$
into $M$, as in $(*)_{2}$ we have ${\rm Rang}(f)\subseteq M_{\ell(1),\ell(2)}$
for some $\\{\ell(1),\ell(2)\\}\in{\mathscr{W}}$, contradiction.]

  1. $(*)_{5}$

$T_{k,n}$ (and ${\rm Mod}_{T^{0}_{\eta,\bar{k}}}$) has the
$(\eta,\bar{k})$-olive property as witnessed by
$\varphi(x_{0},x_{1})=P(x_{0},x_{1}),\psi_{\iota}(x_{0},\dotsc,x_{k(\iota)})=Q_{\iota}(x_{0},\dotsc,x_{k(\iota)})$.

[Why? In Definition 1.1(3), clause (a) holds trivially and clause (c) is
obvious from the choice of $T^{0}_{\eta,\bar{k}}$. For clause $(b)_{\lambda}$
we are given $\langle f_{\alpha}:\alpha<\lambda\rangle$ with $f_{\alpha}$ a
function from $\alpha$ to $\\{0,1\\}$ and we have to find $M$ as there. We
define a $\tau$-model $M$ with:

  1. $\bullet$

universe $\\{a^{*}_{\alpha}:\alpha<\lambda\\}$ such that
$\alpha<\beta\Rightarrow a^{*}_{\alpha}\neq a^{*}_{\beta}$

  2. $\bullet$

$P^{M}=\\{(a^{*}_{\alpha},a^{*}_{\beta}):\alpha<\beta<\lambda\\}$

  3. $\bullet$

$Q^{M}_{\iota}=\\{(a^{*}_{\alpha_{0}},\dotsc,a^{*}_{\alpha_{k(\iota)-1}},a_{\beta}):\alpha_{0}<\ldots<\alpha_{k(\iota)-1}<\beta$
and $f_{\beta}{\restriction}[\alpha_{0},\alpha_{k(\iota)-1}]$ is constantly
$\iota\\}$.

It suffices to prove that $M$ is a model of $T^{0}_{\eta,\bar{k}}$. So toward
a contradiction assume $h$ embeds $N^{*}_{\eta,\bar{k}}$ into $M$, so let
$h(a^{*}_{\ell})=a_{g(\ell)}$ where $g:\\{0,\dotsc,n\\}\rightarrow\lambda$;
necessarily $g$ is a one-to-one function. For $\ell<n$, recall
$N^{*}_{\eta,\bar{k}}\models``P(a^{*}_{\ell},a^{*}_{\ell+1})"$ but $h$ is an
embedding so $M\models``P[a^{*}_{g(\ell)},a^{*}_{g(\ell+1)}]"$, but if
$g(\ell)\geq g(\ell+1)$ this fails by the choice of $P^{M}$ hence
$g(\ell)<g(\ell+1)$. Now let $i_{*}=\min\\{i:\eta(i)=1\\}$. Let
$i_{0}<\ldots<i_{k(0)-1}$ be from $\eta^{-1}\\{0\\}$ such that $i_{0}=0$ and
$i_{k(\iota)-1}$ is maximal hence $i_{*}\in[i_{0},i_{k(0)-1})$. Now
$N^{*}_{\eta,\bar{k}}\models
Q_{0}[a^{*}_{i_{0}},\dotsc,a^{*}_{i_{k(0)-1}},a^{*}_{n}]$ hence $M\models
Q_{0}[a^{*}_{g(i_{0})},\dotsc,a^{*}_{g(i_{k(0)-1})},a^{*}_{g(n)}]$ and this
implies that $f_{g(n)}{\restriction}[g(i_{0}),g(i_{k(0)-1)})]$ is constantly
$0$ hence $f_{g(n)}(g(i_{*}))=0$. Similarly let $j_{0}<\ldots<j_{k(1)-1}$ be
from $\eta^{-1}\\{1\\}$ such that $j_{0}=i_{*}$; now
$N^{*}_{\eta,\bar{k}}\models
Q_{1}[a^{*}_{j_{0}},\dotsc,a^{*}_{j_{k(1)-1}},a^{*}_{n}]$ hence $M\models
Q_{1}[a_{g(0)},\dotsc,a_{g(j_{k(1)-1})},a_{n}]$ hence
$f_{g(n)}{\restriction}[g(j_{0}),g(j_{k(1)-1})]$ is constantly 1, hence
$f_{g(n)}(g(i_{*}))=1$, contradiction. ∎

As in earlier cases we apply a kind of guessing of clubs (almost suitable also
for them i.e. for the proof with strict order and ${\rm SOP}_{4}$). An
unexpected gain here is that here we use a weaker version: there is no
requirement
$\alpha<\lambda\Rightarrow\lambda>|\\{C_{\delta}\cap\alpha:\delta\in S$
satisfies $\alpha\in{\rm nacc}(C_{\delta})\\}|$ but not clear how this helps.
Also here the use of the pair $(\bar{{\mathscr{A}}},\bar{\mathbb{g}})$ may be
helpful.

######  Definition 1.6.

1) For $\lambda$ regular uncountable and $\chi_{2}>\chi_{1}\geq\lambda$ let
${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ mean that there are
$S,\bar{C},I,\bar{{\mathscr{A}}},\bar{\mathbb{g}}$ witnessing it, this means
(note: if $\chi_{1}=\lambda$ then $I=\\{S\\}$):

  1. $\boxplus$

$(a)\quad S\subseteq\lambda$ and $I$ an ideal on $S$

  2. $(b)\quad\bar{C}=\langle C_{\delta}:\delta\in S\rangle$

  3. $(c)\quad C_{\delta}\subseteq\delta$, note that possibly $\sup(C_{\delta})<\delta$

  4. $(d)(\alpha)\quad\bar{\mathbb{g}}=\langle\bar{g}_{j}:j<\chi_{2}\rangle$

  5. $\hskip 10.0pt(\beta)\quad\bar{g}_{j}=\langle g_{j,\delta}:\delta\in S\rangle$

  6. $\hskip 10.0pt(\gamma)\quad g_{j,\delta}:C_{\delta}\rightarrow\\{0,1\\}$

  7. $(f)(\alpha)\quad\bar{{\mathscr{A}}}=\langle\bar{{\mathscr{A}}}_{j}:j<\chi_{2}\rangle$

  8. $\hskip 10.0pt(\beta)\quad\bar{{\mathscr{A}}}_{j}=\langle{\mathscr{A}}_{j,\delta}:\delta\in S\rangle$

  9. $\hskip 10.0pt(\gamma)\quad{\mathscr{A}}_{j,\delta}\subseteq{\mathscr{P}}({\rm nacc}(C_{\delta}))$

  10. $(g)\quad\mathbb{U}_{I}(\chi_{1})<\chi_{2}$; see Definition 0.14, if $\chi_{1}=\lambda$ then we stipulate

$\mathbb{U}_{I}(\chi_{1})=\chi_{1}$ hence this means $\chi_{1}<\chi_{2}$

  11. $(h)\quad$ if $j_{1}\neq j_{2},\delta\in S,A_{1}\in{\mathscr{A}}_{j_{1},\delta}$ and $A_{2}\in{\mathscr{A}}_{j_{2},\delta}$ then there is $\gamma\in A_{1}\cap A_{2}$

such that $g_{j_{1},\delta}(\gamma)\neq g_{j_{2},\delta}(\gamma)$

  12. $(i)\quad$ if $j<\chi_{2}$ and $E$ is a club of $\lambda$ then for some $Y\in I^{+}$ hence $Y\subseteq S$

for every $\delta\in Y$ we have ${\rm nacc}(C_{\delta})\cap
E\in{\mathscr{A}}_{j,\delta}$.

2) For $\ell=1,2,3$ let ${\rm Qr}_{\ell}(\chi_{2},\chi_{1},\lambda)$ be
defined by:

  1. $\bullet$

if $\ell=1$ as above

  2. $\bullet$

if $\ell=2$ as above but there is a sequence $\langle J_{\delta}:\delta\in
S\rangle$ of ideals on ${\rm nacc}(C_{\delta})$ such that
${\mathscr{A}}_{j,\delta}=\\{{\rm nacc}(C_{\delta})\backslash X:X\in
J_{\delta}\\}$

  3. $\bullet$

if $\ell=3$ we use clauses (a)-(g) from part (1) and

  4. $(h)^{-}\quad$ if $E_{j}$ is a club of $\lambda$ for $j<\chi_{2}$ and $\langle\xi_{j}:j<\chi_{2}\rangle$ is a sequence of

ordinals with $\sup\\{\xi_{j}:j<\chi_{2}\\}<\chi_{2}$ then we can find

$j_{1}<j_{2}<\chi_{2},\delta\in S$ and $\gamma\in{\rm nacc}(C_{\delta})$ such
that

$\xi_{j_{1}}=\xi_{j_{2}},\gamma\in E_{j_{1}}\cap E_{j_{2}}$ and
$g_{j_{1},\delta}(\gamma)\neq g_{j_{2},\delta}(\gamma)$.

3) ${\rm Qr}_{\ell,\iota}(\chi_{2},\chi_{1},\lambda)$ is defined as in ${\rm
Qr}_{\ell}(\chi_{2},\chi_{1},\lambda)$ but $g_{j,\delta}:{\rm
nacc}(C_{\delta})\rightarrow\iota_{*}$, etc.

######  Remark 1.7.

Can we weaken the conclusion of clause (i), etc. to:

  1. $\bullet$

$\\{\alpha\in{\rm nacc}(C_{\delta}):\sup(\alpha\cap
E)>\max(C_{\delta}\cap\alpha)\\}\in{\mathscr{A}}_{j,\delta}$.

That is, this suffices in 1.9 but there is no clear gain so have not looked
into it.

######  Fact 1.8.

1) ${\rm Qr}_{2}(\chi_{2},\chi_{1},\lambda)\Rightarrow{\rm
Qr}_{1}(\chi_{2},\chi_{1},\lambda)\Rightarrow{\rm
Qr}_{3}(\chi_{2},\chi_{1},\lambda)$.

2) We have ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ and even ${\rm
Qr}_{2}(\chi_{2},\chi_{1},\lambda)$ when

  1. $(a)$

$\kappa^{+}<\lambda\leq\chi_{1}<\chi_{2}<2^{\kappa}$

  2. $(b)$

$\kappa={\rm cf}(\kappa),\lambda={\rm cf}(\lambda)$

  3. $(c)$

$\mathbb{U}_{I}(\chi_{1})<\chi_{2},I$ an ideal on $S$ so $S\notin I$

  4. $(d)$

$S\subseteq S^{\lambda}_{\kappa}$ is stationary, $\bar{C}=\langle
C_{\delta}:\delta\in S\rangle$ guess clubs, $C_{\delta}\subseteq\delta,{\rm
otp}(C_{\delta})=\kappa$

  5. $(e)$

$I=\\{A\subseteq S$: for some club $E$ of $\lambda$ for no $\delta\in S$ do we
have ${\rm nacc}(C_{\delta})\cap E\in J^{{\rm bd}}_{C_{\delta}}\\}$

###### Proof..

1) Easy.

2) Clause (d) follows by [Sh:420, §2]. The proof itself is straightforward. ∎

######  Theorem 1.9.

1) If $T$ is complete, with the $(\eta,\bar{k})$-olive property and
$\lambda>\kappa^{+}$ and $\lambda,\kappa$ are regular,
$2^{\kappa}>\lambda\geq\kappa^{++}+|T|$ then $T$ has no universal in $\lambda$
(for $\prec$).

2) If $T$ is complete, with the $(\eta,\bar{k},m)$-olive property and
$\lambda={\rm cf}(\lambda)\geq|T|$ and ${\rm
Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ then ${\rm
univ}(\chi_{1},\lambda,T)\geq\chi_{2}$.

3) Similarly for a.e.c. see 1.3(2), so e.g. for universal $K$ with the ${\rm
JEP}$ and the $\lambda-({\rm qf},\eta,\bar{k})$-olive property.

4) We can weaken ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ to ${\rm
Qr}_{1,\theta}(\chi_{2},\chi_{1},\lambda)$ when
$\theta=2^{\partial},\chi_{1}=\chi^{\partial}_{1},\partial<\lambda$.

######  Remark 1.10.

1) We can use ${\rm Qr}_{3}$ instead of ${\rm Qr}_{1}$ by the same proof but
the gain is not clear.

2) If e.g.
$\lambda=\mu^{+},\mu=\mu^{<\mu}=2^{\partial},\chi_{1}=\lambda=\chi_{2}$ (so
have a universal in $\lambda$), failure of ${\rm
Qr}_{1}(\lambda,\lambda,\lambda)$ implies: there is
${\mathscr{F}}\subseteq{}^{\mu}\mu$ such that
$(\forall\eta\in{}^{\mu}\mu)(\exists\nu\in{\mathscr{F}})(\exists^{\mu}i<\mu)(\eta(i)=\nu(i))$.

###### Proof..

1) It follows from (2) by 1.8(2).

2) Let $(\bar{\psi}_{0},\bar{\psi}_{1})$, i.e.
$\bar{\psi}_{\iota}=\langle\psi_{\iota,k}(\bar{x}_{0},\dotsc,\bar{x}_{k}):k=1,\dotsc,k_{\iota}\rangle$
for $\iota=0,1$ witness the $(\eta,\bar{k},m)$-olive property. For simplicity
we can, without loss of generality assume that $m=1$ and $T$ has elimination
of quantifiers and only predicates and its vocabulary is finite. Let
$S,\bar{C},\bar{{\mathscr{A}}},\bar{\mathbb{g}}$ witness ${\rm
Qr}_{1}(\chi_{2},\chi_{1},\lambda)$. For each $j<\chi_{2}$ we define
$\bar{f}_{j}$ by:

  1. $(*)_{1}$

$(a)\quad\bar{f}_{j}=\langle f_{j,\alpha}:\alpha<\lambda\rangle$

  2. $(b)\quad f_{j,\alpha}:\alpha\rightarrow\\{0,1\\}$ is defined by:

    1. $(\alpha)\quad$ if $\beta<\alpha\in S$ then $f_{j,\alpha}(\beta)=g_{j,\alpha}(\min(C_{\alpha}\backslash\beta))$

    2. $(\beta)\quad$ if $\beta<\alpha\in\lambda\backslash S$ then $f_{j,\alpha}(\beta)=0$.

For each $j<\chi_{2}$ we can find $M_{j}\models T$ of cardinality $\lambda$
and pairwise distinct elements $\langle a_{j,\alpha}:\alpha<\lambda\rangle$
satisfying $(b)_{\lambda}$ of Definition 1.1 for $\bar{f}_{j}$. Let
$M_{j,\alpha}=M_{j}{\restriction}\cup\\{a_{j,\beta}:\beta<\alpha\\}$. Let the
function $h^{0}_{j}:\lambda\rightarrow M_{j}$ be defined by
$h^{0}_{j}(\alpha)=a_{j,\alpha}$.

Let ${\mathscr{P}}\subseteq[\chi_{1}]^{\lambda}$ witness
$\mathbb{U}_{J}(\chi_{1})<\chi_{2}$ and for $u\in{\mathscr{P}}$ or just
$u\in[\chi_{1}]^{\lambda}$ let $h^{1}_{u}$ be one to one from $u$ onto
$\lambda$.

Toward contradiction assume that there are $\xi_{*}<\chi_{2}$ and a sequence
$\langle{\mathfrak{A}}_{\xi}:\xi<\xi_{*}\rangle$ of models of $T$ each of
cardinality $\leq\chi_{1}$ witnessing ${\rm
univ}(\chi_{1},\lambda,T)<\chi_{2}$, even equal to $|\xi_{*}|$. Without loss
of generality the universe of each ${\mathfrak{A}}_{\xi}$ is
$\alpha_{\xi}\leq\chi_{1}$. So for every $j<\chi_{2}$ there are
$\xi=\xi_{j}<\xi_{*}$ and an (elementary) embedding $h^{2}_{j}$ of $M_{j}$
into ${\mathfrak{A}}_{\xi}$, hence there is $u_{j}\in{\mathscr{P}}$ such that
$W_{j}:=\\{\alpha\in S:h^{2}_{j}(a_{j,\alpha})\in u_{j}\\}\in I^{+}$ and let
$v_{j}\supseteq u_{j}\cup{\rm Rang}(h^{2}_{j})$ be such that
$v_{j}\in[\chi_{1}]^{\lambda}$ and
${\mathfrak{A}}_{j}{\restriction}v_{j}\prec{\mathfrak{A}}_{j}$ and let
$\langle\gamma_{j,\alpha}:\alpha<\lambda\rangle$ list the members of $v_{j}$.

Let $h_{j}=h^{1}_{v_{j}}\circ h^{2}_{j}\circ(h^{0}_{j}{\restriction}W_{j})$ so
a function from $W_{j}$ into $\lambda$. Let
$N_{j}=({\mathfrak{A}}_{\xi_{j}}{\restriction}v_{j},P^{N_{j}}_{*})$ be the
expansion of ${\mathfrak{A}}_{\xi_{j}}{\restriction}v_{j}$ by the relation
$P^{N_{j}}_{*}={\rm Rang}(h_{j})$ and let $E_{j}=\\{\delta<\lambda:\delta$ is
a limit ordinal,
$(\forall\alpha<\lambda)(h^{1}_{v_{j}}(\alpha)\in\\{\gamma_{j,\beta}:\beta<\delta\\}\equiv\alpha<\delta)$
and $N_{j}{\restriction}\\{\gamma_{j,\alpha}:\alpha<\delta\\}\prec N_{j}\\}$,
clearly a club of $\lambda$. Hence by clause (i) of Definition 1.6(1) there is
$\delta_{j}\in E_{j}\cap S$ such that $A_{j}:={\rm nacc}(C_{\delta})\cap
E_{j}$ belongs to ${\mathscr{A}}_{j,\delta}$.

As $\xi_{*}<\chi_{2},|{\mathscr{P}}|<\chi_{2}$ and
$|\\{h_{j}(a_{j,\delta}):j<\chi_{2},\delta\in
S\\}|<\sup\\{\|{\mathfrak{A}}_{\xi}\|:\xi<\xi_{*}\\}\leq\chi_{1}<\chi_{2}$ by
the pigeon hull principle there are:

  1. $(*)_{2}$

$(a)\quad j_{1}=j(1)<j_{2}=j(2)$

  2. $(b)\quad\xi_{j(1)}=\xi_{j(2)}$

  3. $(c)\quad\delta_{j_{1}}=\delta_{j_{2}}$ call it $\delta$ (so $\delta\in S$)

  4. $(d)\quad u_{j_{1}}=u_{j_{2}}$ call it $u$, so $u=u_{j_{\iota}}\subseteq|N_{j_{\iota}}|$ for $\iota=0,1$

  5. $(e)\quad h_{j_{1}}(a_{j_{1},\delta})=h_{j_{2}}(a_{j_{2},\delta})$ call it $b$, so $b\in{\rm Rang}(h^{2}_{j_{1}})\cap{\rm Rang}(h^{2}_{j_{2}})$.

By clause (h) of Definition 1.6(1) there is $\gamma\in A_{j_{1}}\cap
A_{j_{2}}$ such that $g_{j_{1},\delta}(\gamma)\neq g_{j_{2},\delta}(\gamma)$.

Now we shall choose $\alpha_{\ell}$ by induction on $\ell<n$ such that:

  1. $(*)_{3}$

$(a)\quad\alpha_{\ell}\in W_{j_{\eta(\ell)}}$

  2. $(b)\quad\alpha_{\ell}<\gamma$ but $\alpha_{\ell}>\sup(C_{\delta}\cap\gamma_{\ell})$

  3. $(c)\quad\langle\alpha_{0},\dotsc,\alpha_{\ell}\rangle$ is increasing

  4. $(d)\quad$ in the model $N_{j_{\eta(\ell)+1}}$ the elements $h^{2}_{j_{\eta(\ell)+1}}(a_{j_{1},\delta})=b,h^{1}_{j_{\eta(\ell)+1}}(a_{j_{1},\alpha_{\ell}})$

realize the same quantifier type over
$\\{h_{j_{\eta(\ell(1))+1}}(a_{j_{\eta},\alpha_{\ell(1)}})$:

$\ell(1)<\ell\\}$ or at least for all relevant (finitely many) formulas.

If we succeed, then in the model ${\mathfrak{A}}_{\xi_{*}}$ which extends
$N_{j_{1}}$ and $N_{j_{2}}$ the sequence $\langle
h^{2}_{j_{\eta(\ell)}}(a_{j_{\eta(\ell)},\alpha_{\ell}}):\ell<n\rangle\char
94\relax\langle b\rangle$ realizes the “forbidden” type that is the one from
clause (c) of Definition 1.1, contradiction.

As $\delta\in W_{j}\cap E_{j_{\eta(\ell)}}$ by the choice of
$E_{j_{\eta(\ell)}}$ we can carry the induction.

3) similarly.

4) As in [Sh:457] and the above, just use $\partial$-tuples of $\bar{a}$’s. ∎

A sufficient condition for cases of ${\rm Qr}_{i}$ is

######  Definition 1.11.

Let ${\rm Qr}_{4}(\lambda)$ mean: $\lambda=\mu^{+}$ and $\langle
C_{\delta},D_{\delta}:\delta\in S\rangle$ satisfies
$C_{\delta}\subseteq\delta,D_{\delta}$ a filter on ${\rm nacc}(C_{\delta})$
such that ${\mathscr{P}}({\rm nacc}(C_{\delta}))/D_{\delta}$ satisfies the
$2^{\mu}$-c.c. and for every club $E$ of $\lambda$ for some $\delta\in
S,E\cap{\rm nacc}(C_{\delta})\in D^{+}_{\delta}$.

##  2\. The class of Groups have the olive property

We shall try to prove that the class of groups has a universal member almost
only when cardinal arithmetic is close to G.C.H. This is done by

######  Theorem 2.1.

The class of groups has the olive property, see Definition 0.8 or
1.1(1)(d)$(\gamma)$, in fact, the $(\eta,\bar{k},m)$-olive property, where
$\eta=\langle 0,1,0\rangle,\bar{k}=(2,1),m=6$.

We break the proof into a series of definitions and claims; we may replace the
use of HNN extension (in 2.14) and free amalgamation (in 2.13) by the proof of
2.15.

######  Definition 2.2.

Let $\bar{\psi}=\bar{\psi}^{{\rm grp}}_{{\rm olive}}$ be
$(\varphi_{0,1},\varphi_{0,2},\varphi_{1,1})$ defined as follows (letting
$m=6$):

  1. $(a)$

$\psi_{0,1}=\varphi_{0}=\varphi_{0}(\bar{x}_{[m]},\bar{y}_{[m]})=y^{-1}_{5}x_{0}y_{5}=x_{2}$

  2. $(b)$

$\psi_{1,1}=\varphi_{1}=\varphi_{1}(\bar{x}_{[m]},\bar{y}_{[m]})=x^{-1}_{5}y_{1}x_{5}=y_{3}\wedge
x^{-1}_{5}y_{4}x_{5}=y_{4}$

  3. $(c)$

$\psi_{0,2}=\psi(\bar{x}_{[m]},\bar{y}_{[m]},\bar{z}_{[m]})=\sigma_{*}(x_{0},y_{1},z_{4})=e\wedge\sigma_{*}(x_{2},y_{3},z_{4})\neq
e$, on $\sigma_{*}$ see below.

######  Definition/Claim 2.3.

There is a $\sigma_{*}=\sigma_{*}(x,y,z)$ such that:

  1. $(a)$

$\sigma_{*}$ is a group word

  2. $(b)$

for some group $G$ and $a,b,c\in G$ we have “$\sigma_{*}(a,b,c)\neq e_{G}$”

  3. $(c)$

for any group $G$ and $a,b,c\in G$ we have
$e\in\\{a,b,c\\}\Rightarrow\sigma^{G}(a,b,c)=e_{G}$

######  Remark 2.4.

Earlier we intend to use [LS77] hence add

  1. $(d)$

for no two distinct interval $\sigma_{1},\sigma_{2}$ of some cyclic
permutation $\sigma^{\prime}$ of $\sigma$ do we have
$\sigma_{2}\in\\{\sigma_{1},\sigma^{-1}_{2}\\}$ and $\ell g(\sigma_{1})>\ell
g(\sigma)/6$; it seems not necessary

  2. $(e)$

$(\alpha)\quad\sigma_{*}$ is cyclically reduced.

But this is not necessary.

###### Proof..

Straightforward, e.g. $(x^{-1}y^{-1}x^{-1}y)^{-1}z^{-1}(x^{-1}y^{-1}xy)z$. ∎

######  Claim 2.5.

The $\bar{\psi}$ from 2.2 satisfies clause (c) of Definition 0.8 or 1.1(3),
i.e. for no group $G$ and $\bar{a}_{\ell}\in{}^{m}G$ for $\ell<4$ do the
formulas there hold.

######  Remark 2.6.

We prove more: there are no group $G$ and $\bar{a}_{\ell}\in{}^{m}G$ for
$\ell=0,1,2,3$ such that
$\varphi_{0}[\bar{a}_{0},\bar{a}_{1}],\varphi_{1}[\bar{a}_{1},\bar{a}_{2}],\varphi_{1}[\bar{a}_{1},\bar{a}_{3}]$
and $\psi[\bar{a}_{0},\bar{a}_{2},\bar{a}_{3}]$.

###### Proof..

Assume toward contradiction that $G,\langle\bar{a}_{\ell}:\ell<4\rangle$ forms
a counterexample; now conjugation by $a_{1,5}$ is an automorphism of $G$ which
we call $g$.

Now:

  1. $\bullet$

$g(a_{0,0})=a_{0,2}$ by (a) of 2.2 as
$G\models\varphi_{0}[\bar{a}_{0},\bar{a}_{1}]$

  2. $\bullet$

$g(a_{2,1})=a_{2,3}$ by first conjunct of (b) of 2.2 as
$G\models\varphi_{1}[\bar{a}_{1},\bar{a}_{2}]$

  3. $\bullet$

$g(a_{3,4})=a_{3,4}$ by the second conjunct of (b) of 2.2 as
$G\models\varphi_{1}[\bar{a}_{1},\bar{a}_{3}]$.

Together

  1. $\bullet$

$g(\sigma_{*}(a_{0,0},a_{2,1},a_{3,4}))=\sigma_{*}(a_{0,2},a_{2,3},a_{3,4})$

but this contradicts $G\models\psi[\bar{a}_{0},\bar{a}_{2},\bar{a}_{3}]$, see
clause (c) of 2.2. ∎

######  Definition 2.7.

Let $\bar{f}\in\mathbb{F}_{\lambda}$, i.e. $\bar{f}=\langle
f_{\alpha}:\alpha<\lambda\rangle,f_{\alpha}:\alpha\rightarrow\\{0,1\\}$.

1) Let $X_{\bar{f}}=X_{\bar{f},m}$ where we let
$X_{\bar{f},k}=\\{x_{\alpha,\ell}:\alpha<\lambda,\ell<k\\}$ for $k\leq m$;
recall that here $m=6$.

2) Let $\bar{x}_{\alpha,k}=\langle x_{\alpha,\ell}:\ell<k\rangle$ for $k\leq
m$ and let $\bar{x}_{\alpha}=\bar{x}_{\alpha,m}$.

3) For $\ell=0,1$ we define the set $\Gamma^{\ell}_{\bar{f}}$ of equations
(pedantically, for $\ell=0$ conjunctions of two equations):

$\\{\varphi_{\ell}(\bar{x}_{\alpha},\bar{x}_{\beta}):\alpha<\beta<\lambda\text{
and }f_{\beta}(\alpha)=\ell\\}.$

4) We define the set $\Gamma^{2}_{\bar{f}}$ of equations

$\\{\sigma_{*}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})=e:\alpha<\beta<\gamma<\lambda\text{
and }f_{\gamma}{\restriction}[\alpha,\beta]\text{ is constantly }0\\}.$

5) Let $G^{5}_{\bar{f}}$ be the group generated by $X_{\bar{f},5}$ freely
except the equations in $\Gamma^{2}_{\bar{f}}$, note that the $x_{\alpha,5}$’s
are not mentioned in $\Gamma^{2}_{\bar{f}}$.

6) Let $G^{6}_{\bar{f}}$ be the group generated by $X_{\bar{f},6}$ freely
except the equations in
$\Gamma^{0}_{\bar{f}}\cup\Gamma^{1}_{\bar{f}}\cup\Gamma^{2}_{\bar{f}}$.

######  Discussion 2.8.

For our purpose we have to show that for $\alpha<\beta<\gamma$ (and
$\bar{f}\in\mathbb{F}_{\lambda}$) we have:
$G_{\bar{f},6}\models``\psi[\bar{x}_{\alpha},\bar{x}_{\beta},\bar{x}_{\gamma}]"$
iff $f_{\gamma}{\restriction}[\alpha,\beta]=0_{[\alpha,\beta]}$. For proving
the “if” implication, assume
$f_{\gamma}{\restriction}[\alpha,\beta]=0_{[\alpha,\beta]}$. Now the
satisfaction of “$\sigma_{*}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})=e"$ is
obvious by the role of $\Gamma^{2}_{\bar{f}}$, the analysis below is intended
to prove the other half
“$\sigma_{*}(x_{\alpha,2},x_{\beta,3},x_{\gamma,4})\neq e$”. For proving the
“only if” implication it suffices to prove that
“$\sigma_{*}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})\neq e"$ when
$f_{\gamma}{\restriction}[\alpha,\beta]\neq 0_{[\alpha,\beta]}$. For both
cases, we prove that this holds in $G^{5}_{\bar{f}}$ and then prove that
$G^{5}_{\bar{f}}\subseteq G^{6}_{\bar{f}}$ in the natural way.

######  Claim 2.9.

1) If $\alpha<\beta<\gamma<\lambda$ and
$f_{\gamma}{\restriction}[\alpha,\beta]\neq 0_{[\alpha,\beta]}$ then
$G^{5}_{\bar{f}}\models``\sigma_{*}[x_{\alpha,0},x_{\beta,1},x_{\gamma,4}]\neq
e"$.

2) If $\alpha<\beta<\gamma<\lambda$ then
$G^{5}_{\bar{f}}\models``\sigma_{*}(x_{\alpha,2},x_{\beta,3},x_{\gamma,4})\neq
e"$.

###### Proof..

1) Use 2.10 below with $X=\\{X_{\xi,\ell}:\xi\in\\{\alpha,\beta,\gamma\\}$ and
$\ell<5\\}$.

2) Use 2.10(2) below with $X=\\{x_{\xi,\ell}:\xi<\lambda,\ell<5$ and
$\ell>0\\}$. ∎

######  Observation 2.10.

1) If $x_{\alpha,\ell},x_{\beta,k}\in X^{5}_{\bar{f}}$ and
$(\alpha,\ell)\neq(\beta,k)$ then $G^{5}_{\bar{f}}\models``x_{\alpha,\ell}\neq
x_{\beta,k}"$.

2) If $X\subseteq X^{5}_{\bar{f}}$ and
$(\sigma_{*}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})=e)\in\Gamma^{2}_{\bar{f}}\Rightarrow\\{x_{\alpha,0},x_{\beta,1},x_{\gamma,4}\\}\nsubseteq
X$ then $X$ generates freely a subgroup of $G^{5}_{\bar{f}}$.

###### Proof..

1) Let $G^{\prime}=\oplus\\{{\mathbb{Z}}x:x\in X^{5}_{\bar{f}}\\}$, it is an
abelian group; let
$G^{\prime\prime}=\oplus\\{{\mathbb{Z}}x_{\alpha,i}:\alpha<\lambda,i\notin\\{\ell,k\\}\\}$
a subgroup. So $G^{\prime}/G^{\prime\prime}$ by clause (c) of Definition 2.3,
satisfies all the equations in $\Gamma^{2}_{\bar{f}}$ and it satisfies the
desired inequality. As $G^{5}_{\bar{f}}$ is generated by $X^{5}_{\bar{f}}$
freely except the equations in $\Gamma^{2}_{\bar{f}}$ the desired result
follows. Alternatively use part (2).

2) Let $H=H_{X}$ be the group generated by $X$ freely. We define a function
$F$ from $X^{5}_{\bar{f}}$ into $H$ by

  1. $\bullet$

$F(x)$ is $x$ if $x\in X$ and is $e_{H}$ if $x\in X^{5}_{\bar{f}}\backslash
X$.

Now $F$ respects every equation form $\Gamma^{2}_{\bar{f}}$ by clause (c) of
2.5, hence $f$ induces a homomorphism from $G^{5}_{\bar{f}}$ into $H$, really
onto. Hence the desired conclusion follows. ∎

######  Definition 2.11.

For $\beta<\lambda$ we define a partial function $F_{\beta}$ from
$X^{5}_{\bar{f}}$ to $X^{5}_{\bar{f}}$:

  1. $\bullet$

if $\alpha<\beta$ and $f_{\beta}(\alpha)=0$ then
$F_{\beta}(x_{\alpha,0})=x_{\alpha,2}$

  2. $\bullet$

if $\gamma>\beta$ and $f_{\gamma}(\beta)=1$ then
$F_{\beta}(x_{\gamma,1})=x_{\gamma,3},F_{\beta}(x_{\gamma,4})=x_{\gamma,4}$.

######  Claim 2.12.

1) $F_{\beta}$ is a well defined partial one-to-one function from
$X^{5}_{\bar{f}}$ to $X^{5}_{\bar{f}}$.

2) The domain and the range of $F_{\beta}$ satisfies the criterion of 2.10(2).

###### Proof..

1) It is a function as no $x_{\alpha,\ell}$ appears in two cases. Also if
$F_{\beta}(x_{\alpha_{1},\ell})=x_{\alpha_{2},k}$ then
$\alpha_{1}=\alpha_{2}\wedge(\ell,k)\in\\{(0,2),(1,3),(4,4)\\}$ so $F_{\beta}$
is one to one.

2) Assume
$[\sigma_{*}(x_{\alpha_{1},0},x_{\alpha_{2},1},x_{\alpha_{3},4})=e]\in\Gamma^{2}_{\bar{f}}$
so

  1. $(*)_{1}$

$\alpha_{1}<\alpha_{2}<\alpha_{3}$
and

  2. $(*)_{2}$

$f_{\alpha,3}{\restriction}[\alpha_{1},\alpha_{2}]=0_{[\alpha_{1},\alpha_{2}]}$.

First, toward contradiction assume
$\\{x_{\alpha_{1},0},x_{\alpha_{2},1},x_{\alpha_{3},4}\\}\subseteq{\rm
Dom}(F_{\beta})$.

Now if $\alpha_{1}\geq\beta$ then $x_{\alpha_{1},0}\notin{\rm
Dom}(F_{\beta})$, just inspect Definition 2.11 so necessarily
$\alpha_{1}<\beta$ and similarly $f_{\beta}(\alpha_{1})=0$ (but not used).

If $\alpha_{2}\leq\beta$ then $x_{\alpha_{2},1}\notin{\rm Dom}(F_{\beta})$, so
$\beta<\alpha_{2}$ and similarly $f_{\alpha_{2}}(\beta)=1$ (again not used) so
together $\alpha_{1}<\beta<\alpha_{2}$. Also as $x_{\alpha_{3},4}\in{\rm
Dom}(F_{\beta})$ it follows that ($\beta<\alpha_{3}$ which follows by earlier
inequalities and) $f_{\alpha_{3}}(\beta)=1$, so together $\beta$ witness that
$f_{\alpha_{3}}{\restriction}[\alpha_{1},\alpha_{2}]$ is not constantly zero
contradiction to
$[\sigma(x_{\alpha_{1},0},x_{\alpha_{2},0},x_{\alpha_{3},0})=e]\in\Gamma^{2}_{\bar{f}}$.

Second, toward contradiction assume
$\\{x_{\alpha_{2},0},x_{\alpha_{1},2},x_{\alpha_{3},4}\\}\subseteq
X\subseteq{\rm Rang}(F_{\beta})$, but “$x_{\alpha_{2},0}\in{\rm
Rang}(F_{\beta})"$ is impossible by Definition 2.11. ∎

######  Claim 2.13.

To prove $G^{5}_{\bar{f}}\subseteq G^{6}_{\bar{f}}$ any of the following
conditions suffice:

  1. $(a)$

there are a group $H$ extending $G^{5}_{\bar{f}}$ and $y_{\zeta}\in G$ for
$\zeta<\lambda$ such that $\zeta<\lambda\wedge
F_{\zeta}(x_{\varepsilon_{1},\ell_{1}})=x_{\varepsilon_{2},\ell_{2}}\Rightarrow
H\models``y^{-1}_{\zeta}x_{\varepsilon_{1},\ell_{1}}y_{\zeta}=x_{\varepsilon_{2},\ell_{2}}"$

  2. $(b)$

for each $\zeta<\lambda$ there is a group $H$ extending $G^{5}_{\bar{f}}$ and
$y\in G$ such that
$F_{\zeta}(x_{\varepsilon_{1},\ell_{1}})=x_{\varepsilon_{2},\ell_{2}}\Rightarrow
H\models``y^{-1}x_{\varepsilon_{1},\ell_{1}}y=x_{\varepsilon_{2},\ell_{2}}"$.

###### Proof..

Clause (a) suffice:

We define a function $F$ from $X^{6}_{\bar{f}}$ into $H$ by:

  1. $\bullet$

$F(x_{\varepsilon,\ell})$ is $x_{\varepsilon,\ell}\in G^{5}_{\bar{f}}\subseteq
H$ if $\ell<5\wedge\varepsilon<\lambda$

is $y_{\zeta}$ if $\varepsilon=\zeta\wedge\ell=5$.

Check that the mapping $F$ respects the equations in
$\Gamma^{0}_{\bar{f}}\cup\Gamma^{1}_{\bar{f}}\cup\Gamma^{2}_{\bar{f}}$ hence
it induces a homomorphism $F^{1}$ from $G^{6}_{\bar{f}}$ into $H$, for every
group word
$\sigma=\sigma(\ldots,x_{\varepsilon_{i},\ell_{i}},\ldots)_{i<n},x_{\varepsilon_{i},\ell_{i}}\in
X^{5}_{\bar{f}}$, we have $G^{6}_{\bar{f}}\models``\sigma=e"\Rightarrow
G^{5}_{\bar{f}}\models``\sigma=e"$, so we are done.

Clause (b) suffice:

Let $(H_{\zeta},y_{\zeta})$ for $\zeta<\lambda$ be as guaranteed by the
assumption, i.e. clause (b). Without loss of generality
$\zeta\neq\xi<\lambda=G_{\zeta}\cap G_{\xi}=G^{5}_{\bar{f}}$. Now clause (a)
follows by using free amalgamation of $\langle H_{\zeta}:\zeta<\lambda\rangle$
over $G^{5}_{\bar{f}}$, we know it is as required in clause (a), see e.g.
[LS77]. ∎

######  Claim 2.14.

1) Clause (b) of 2.13 holds.

2) The conclusion of claim 2.13 holds.

3) The conclusions of 2.9 hold also for $G^{6}_{\bar{f}}$.

###### Proof..

1) By the theorems on HNN extensions see [LS77] applied with the group being
$G^{5}_{\bar{f}}$ and the partial automorphism $\pi_{\zeta}$ being the one
$F_{\zeta}$ induced, i.e.

  1. $\bullet$

${\rm Dom}(\pi_{\zeta})$ is the subgroup of $G^{5}_{\bar{f}}$ generated by
${\rm Dom}(F_{\zeta})$

  2. $\bullet$

$\pi_{\zeta}(x_{\varepsilon,\ell})=F_{\zeta}(x_{\varepsilon,\ell})$ for
$x_{\varepsilon,\ell}\in{\rm Dom}(F_{\zeta})$.

By claim 2.12(2) and 2.10(2) we know that $\pi_{\zeta}$ is indeed an
isomorphism.

2) Follows by 2.13 and 2.14.

3) By 2.9 and part (2). ∎

Proof of 2.1: Should be clear by now.

$*\qquad*\qquad*$

######  Claim 2.15.

The pair $(K_{{\rm lfgr}},K_{{\rm gr}})$ of classes, i.e. (locally finite
groups, groups), has the olive property, as witnessed by $\bar{\varphi}$ from
2.2.

###### Proof..

We rely on observation 2.16 below and use its notation. Let
$J=\\{(\alpha,\beta,\gamma):\alpha<\beta<\gamma<\lambda$ and
$f_{\gamma}{\restriction}[\alpha,\beta]=0_{[\alpha,\beta]}\\}$.

Let $G^{5}_{\bar{f}},G^{6}_{\bar{f}}$ be as in the proof of 2.1, that in
Definition 2.7. Now for $\bar{\alpha}=(\alpha_{0},\alpha_{1},\alpha_{2})\in J$
let $\pi^{5}_{\bar{\alpha}}$ be the function from $X_{\bar{f},5}$ (see
Definition 2.7) into $K$ defined by

  1. $(*)_{1}$

$\pi^{5}_{\bar{\alpha}}(x_{\beta,k})$ is

    1. $\bullet$

$e_{K}$ if $\beta\notin\\{\alpha_{0},\alpha_{1},\alpha_{2}\\}$

    2. $\bullet$

$z_{\ell,k}$ if $\beta=\alpha_{\ell}$.

Now

  1. $(*)_{2}$

$\pi^{5}_{\bar{\alpha}}$ respects the equations from $\Gamma^{2}_{\bar{f}}$.

[Why? The equation
$\sigma_{*}(x_{\alpha_{0},0},x_{\alpha_{1},1},x_{\alpha_{2},4})=e$ holds as
$K$ satisfies $\sigma_{*}(z_{0,0},z_{1,1},z_{2,4})=e$. For the other equations
see 2.3(c).]

Let $\pi^{6}_{\bar{\alpha}}$ be the following function from $X^{6}_{\bar{f}}$
into $K$:

  1. $(*)_{3}$

$\pi^{6}_{\bar{\alpha}}(x)$ is

    1. $\bullet$

$\pi^{5}_{\bar{\alpha}}(x)$ when $x\in X^{5}_{\bar{f}}$

    2. $\bullet$

$z_{s}$ when $x=x_{\beta,5},\beta<\lambda$ and
$s=s_{\bar{\alpha},\beta}:=(\\{\ell\leq 2:\alpha_{\ell}<\beta$ and
$f_{\beta}(\alpha_{\ell})=0\\},\\{\ell\leq 2:\beta<\alpha_{\ell}$ and
$f_{\alpha_{\ell}}(\beta)=1\\})$.

[Why is $\pi^{6}_{\bar{\alpha}}$ as required? The least obvious point is: why
$s\in S_{*}$? Let $s=(u_{1},u_{1})$, now $\ell_{1}\in u_{1}\wedge\ell_{2}\in
u_{2}\Rightarrow\alpha_{\ell_{1}}<\beta<\alpha_{\ell_{2}}\Rightarrow\ell_{1}<\ell_{2}$
and $(\\{0\\},\\{1,2\\})\neq s$ because
$f_{\alpha_{2}}{\restriction}[\alpha_{0},\alpha_{1}]$ is constantly zero.]

  1. $(*)_{4}$

$\pi^{6}_{\bar{\alpha}}$ respects the equations in
$\Gamma^{0}_{\bar{f}}\cup\Gamma^{1}_{\bar{f}}$.

[Why? Check the definitions.]

By $(*)_{2},(*)_{4}$ there is a homomorphism $\pi_{\bar{\alpha}}$ from
$G_{\bar{f}}$ into $K$. Let $G_{*}$ be the product of $J$-copies of $K$, i.e.

  1. $(*)_{5}$

$(a)\quad$ the set of elements of $G_{*}$ is the set of functions $g$ from $J$
into $K$

  2. $(b)\quad G_{*}\models``g_{1}g_{2}=g_{3}"$ iff $\bar{\alpha}\in J\Rightarrow K\models``g_{1}(\bar{\alpha})g_{2}(\bar{\alpha})=g_{3}(\bar{\alpha})"$

  3. $(*)_{6}$

$G_{*}$ is a locally finite group

  4. $(*)_{7}$

for $\alpha<\lambda,k<m$ let $\bar{g}_{\beta}=\langle g_{\beta,k}:k<m\rangle$
where $g^{*}_{\beta,k}\in G_{*}$ be defined by
$(g_{\beta,k}(\bar{\alpha}))(x)=\pi^{6}_{\bar{\alpha}}(x_{\beta,k})$

  5. $(*)_{8}$

$G_{*},\langle\bar{g}_{\beta}:\beta<\lambda\rangle$ witnesses the olive
property.

[Why? Check.]

So we are done. ∎

######  Observation 2.16.

There are $K,z_{i,k}(i<3,k<m)$ and $\langle\pi_{s}:s\in S_{*}\rangle$ such
that:

  1. $(a)$

$K$ is a finite group

  2. $(b)$

$z_{i,k}\in K$

  3. $(c)$

$\sigma_{*}(z_{0,0},z_{1,2},z_{2,4})=e$ but
$\sigma_{*}(z_{0,2},z_{1,3},z_{2,4})\neq e$

  4. $(d)$

$S_{*}=\\{(u_{1},u_{2}):u_{1},u_{2}\subseteq\\{0,1,2\\}$ and
$(\forall\ell_{1}\in u_{1})(\forall\ell_{2}\in u_{2})(\ell_{1}<\ell_{2})$ but
$(u_{1},u_{2})\neq(\\{0\\},\\{1,2\\})$

  5. $(e)$

for $s=(u_{1},u_{2})\in S_{*}$ we have: $\pi_{s}$ is a partial isomorphism of
$K$ such that:

    1. $(\alpha)$

if $\ell\in u_{1}$ then $\pi_{s}(x_{\ell,0})=x_{\ell,2}$

    2. $(\beta)$

if $\ell\in u_{2}$ then
$\pi_{s}(x_{\ell,1})=x_{\ell,2},\pi_{s}(z_{\ell,4})=z_{\ell,4}$

  6. $(f)$

moreover there are $z_{s}\in K$ for $s\in S_{*}$ such that $(\forall x\in{\rm
Dom}(\pi_{s}))(\pi_{s}(x)=z^{-1}_{s}xz_{s})$.

###### Proof..

First, we ignore clause (f). We use finite nilpotent groups. Let
$n_{2}=3m,n_{1}=\binom{n_{2}}{2},n_{0}=\binom{n_{1}}{2}$, let
$f_{\ell}:[n_{\ell+1}]^{2}\rightarrow n_{\ell}$ be one-to-one for $\ell=0,1$.

Let $K_{1}$ be the group generated by $\\{y_{j,\ell}:j\leq 2,\ell<n_{j}\\}$
freely except the equations

  1. $(*)_{1}$

$(a)\quad y_{j,\ell}\cdot y_{j,\ell}=e$

  2. $(b)\quad[y_{j+1,\ell_{1}},y_{j+1,\ell_{2}}]=y_{j,f\\{\ell_{1},\ell_{2}\\}}$, i.e. $y^{-1}_{j+1,\ell_{1}}y^{-1}_{j+1,\ell_{2}}y_{j+1,\ell_{2}}y_{j+1,\ell_{2}}=y_{j,f\\{\ell_{1},\ell_{2}\\}}$

when $j<2,\ell_{1}<\ell_{2}<n_{j+1}$

  3. $(c)\quad[y_{j_{1},\ell_{1}},y_{j_{2},\ell_{2}}]=e$ when $(j_{1}=0=j_{2})\vee(j_{1}\neq j_{2}\leq 2)$ and

$\ell_{1}<n_{j_{1}},\ell_{2}<n_{j_{2}}$.

Clearly $K_{1}$ is finite.

Let $z_{i,\ell}=y_{2,6i+\ell}$ for $i<3,\ell<m$, let $\ell_{*}$ be such that
$[[z_{0,0},z_{1,1}],z_{2,4}]=y_{0,\ell_{*}}$. Let $K_{0}$ be the subgroup
$\\{e,y_{0,\ell_{*}}\\}$ of $K$, it is a normal subgroup as it is included in
the center of $K_{1}$ and let $K_{2}=K_{1}/K_{0}$ and we define $z_{i,\ell}$
as $y_{i,\ell}/K_{0}$.

Now

  1. $(*)_{2}$

$K_{2},\langle z_{i,\ell}:i\leq 2,\ell<m\rangle$ are as required in (a)-(e) of
the claim.

[Why? We should just check that for $s\in S_{*}$ there is $\pi_{s}$ as
required, i.e. that some subgroups of $K_{2}$ generated by subsets of $\langle
z_{i,\ell}:i\leq 2,\ell<m\rangle$ are isomorphic, but as none of them included
$\\{z_{0,0},z_{1,1},z_{2,4}\\}$ and the way $K_{2}$ was defined this is
straightforward.]

Lastly, there is a finite group $K$ extending $K_{2}$ and $z_{s}\in K$ for
$s\in S$ such that $x\in{\rm Dom}(\pi_{s})\Rightarrow
z^{-1}_{s}xz_{s}=\pi_{s}(x)$. Why? Simply $K_{2}$ can be considered as a group
of permutations of the set $K_{2}$ (e.g. multiplying from the right), and it
is easy to find $z_{s}\in{\rm Sym}(K_{2})$ as required. ∎

######  Conclusion 2.17.

Assume ${\rm Qr}_{1}(\chi_{1},\chi_{2},\lambda)$.

Then there is no sequence $\langle G_{\alpha}:\alpha<\alpha_{*}\rangle$ of
length $<\chi_{2}$ of groups of cardinality $\leq\chi_{1}$ such that any
locally finite group $H$ of cardinality $\lambda$ can be embedded into at
least one of them.

So, e.g.

######  Conclusion 2.18.

1) If $\mu={\rm cf}(\mu),\mu^{+}<\lambda={\rm cf}(\lambda)<2^{\mu}$ then there
is no group of cardinality $\lambda$ universal for the class of locally finite
groups.

2) E.g. if $\aleph_{2}\leq\lambda={\rm cf}(\lambda)<2^{\aleph_{0}}$ this
applies.

##  3\. Concluding remarks

We may like to weaken the model theoretic condition but add to the property
${\rm Qr}$ of the relevant cardinals that “the $C_{\delta}$’s has few
branches”. It is not clear whether there will be any gain.

######  Definition 3.1.

$T$ has the $(\eta,\bar{k},m)-*-\Delta$-olive property when
$\Delta\subseteq{\mathbb{L}}(\tau_{T})$ and for some
$(\bar{\varphi}_{1},\bar{\varphi}_{2})$ we have (for every $\lambda$)

  1. $(a)$

for $\iota=0,1$ we have
$\bar{\varphi}_{\iota}=\langle\varphi_{\iota,\ell}(\bar{x}_{0},\dotsc,\bar{x}_{\ell-1}):\ell=2,\dotsc,k_{\iota}\rangle$
with $\varphi_{\iota,\ell}\in\Delta$ and $m=\ell g(\bar{x}_{0})=\ldots\ell
g(\bar{x}_{k-1})$

  2. $(b)_{\lambda}$

for every $I\in K_{{\rm etr}}$ (see 3.4, old $\boxplus_{1}$ of the proof of
LABEL:n2) we can find $\bar{\mathbb{a}}$ such that

    1. $(\alpha)$

$\bar{\mathbb{a}}=\langle\bar{a}_{\eta}:\eta\in P_{I}\rangle$

    2. $(\beta)$

$\bar{a}_{\eta}={}^{m}{\mathfrak{C}}$ where
${\mathfrak{C}}={\mathfrak{C}}_{T}$

    3. $(\gamma)$

if $\iota<2,\ell\in\\{2,\dots,k\\}$ and
$\bar{\eta}=\langle\eta_{0},\dotsc,\eta_{\ell-1}\rangle$ is an
$(I,\iota)$-sequence (i.e. $\eta_{i}\in P_{I},F_{I,\iota}(\eta_{i})<^{{\rm
tr}}_{I}\eta_{i+1}$ (when defined) then
${\mathfrak{C}}\models\varphi_{\iota,\ell}[\bar{a}_{\eta_{0}},\dotsc,\bar{a}_{\eta_{\ell-1}}]$

  3. $(c)$

there are no $\bar{a}_{i}\in{}^{m}{\mathfrak{C}}$ for $\ell<n+1$ such that:

  4. $\bullet\quad$ if $\iota\in\\{0,1\\},\ell\in\\{2,\dotsc,k\\},i_{0}<\ldots<i_{k-1}<n$ and

$\ell<k\Rightarrow\eta(\ell)=\iota$ then
${\mathfrak{C}}\models\varphi_{\iota,\ell}[\bar{a}_{i_{0}},\dotsc,\bar{a}_{i_{\ell-1}},\bar{a}_{n}]$.

######  Definition 3.2.

We say an a.e.c. ${\mathfrak{k}}$ has the $(\eta,\bar{k},<\sigma)$-olive when
: there are pairs of sequences of formulas
$(\bar{\varphi}_{0},\bar{\varphi}_{1})$ which are ${\mathfrak{k}}$-upward
invariant (see 0.13) with $\ell g(x_{\zeta})=\varepsilon<\sigma$ such that for
every $I\in K_{{\rm etr}}$ (see LABEL:n2) of cardinality $\lambda$ there is
$M\in K_{{\mathfrak{k}}}$ of cardinality $\lambda$ and
$\bar{a}_{\eta}\in{}^{\varepsilon}M$ for $\eta\in P_{\eta}$ such that the
parallel of 3.1 holds.

######  Discussion 3.3.

The intention is to have a parallel of §1 with somewhat weaker version of the
olive here, but the price is a somewhat stronger set theoretic condition.

######  Definition 3.4.

1) Let $K_{{\rm etr}}$ (expanded tree) be the class of structures
$I=({\mathscr{T}},<_{{\rm lin}},<_{{\rm
tr}},P,F_{0},F_{1})=({\mathscr{T}}_{I},<^{{\rm
lin}}_{I},P_{I},F_{I,0},F_{I,1})$ satisfying:

  1. $(a)$

${\mathscr{T}}_{I}=({\mathscr{T}},<_{{\rm
tr}})=({\mathscr{T}},\leq_{{\mathscr{T}}})=({\mathscr{T}}_{I},\leq^{{\rm
tr}}_{I})$ is a partial order; moreover a well founded tree

  2. $(b)$

$P\subseteq I$

  3. $(c)$

$F_{I,\ell}$ is a one-to-one function from $P_{I}$ into $I\backslash P_{I}$,
for $\ell=0,1$

  4. $(d)$

${\mathscr{T}}$ is the disjoint union of $P_{I},{\rm Rang}(F_{I,0}),{\rm
Rang}(F_{I,1})$

  5. $(e)$

if $\ell<2$ and $t\in P_{I}$ then $F_{I,\ell}(t)$ is a successor of $t$, i.e.
$t<_{I}F_{I,\ell}(t)$ and $\neg(\exists s\in
I)(t<_{{\mathscr{T}}}s<_{{\mathscr{T}}}F_{{\mathscr{T}},\ell}(t))$

  6. $(f)$

if $t\in P_{I}$ and $t<_{{\mathscr{T}}}s$ then
$\bigvee\limits_{\ell>n}F_{\ell}(t)\leq_{{\mathscr{T}}}s$

  7. $(g)$

$({\mathscr{T}},<_{{\rm lin}})=({\mathscr{T}},<^{{\rm lin}}_{I})$ is a linear
order

  8. $(h)$

if $\ell<2,s\in P_{I}$ and $F_{\ell}(s)\leq_{k}t_{\ell}$ for $\ell=0,1$ then
$t_{0}<_{{\rm lin}}s<_{{\rm lin}}t_{s}$

2) We define $K_{{\rm ftr}}$ as $\\{J_{I}:I\in K_{{\rm etr}}\\}$ where for
$I\in K_{{\rm etr}}$ let $J_{I}$ be the structure $(P_{I},<^{{\rm
tr}}_{I}{\restriction}P_{I},<^{{\rm
lin}}_{I}{\restriction}P_{I},Q_{I,0},Q_{I,1}$ where
$Q^{\iota}_{I}=\\{(\eta,\nu):\eta\in P_{I},\nu\in P_{I}$ and
$F_{I,\iota}(\eta)\leq^{{\rm tr}}_{I}\nu\\}$.

######  Definition 3.5.

1) Let “$T$ have the $(\Delta,\eta,\bar{k},m)$-olive property where
$\eta\in{}^{n}(\iota_{*}),\bar{k}=\langle
k_{\iota}=k(\iota):\iota<\iota_{*}\rangle$, is defined (similarly to
Definition 1.1 but 2 is replaced by $\iota_{*}$ and $\bar{k}=(k_{0},k_{1})$ by
$\bar{k}=\langle k_{\iota}:\iota<\iota_{*}\rangle$) when there are formulas
$\varphi(\bar{x}_{[m],0},\bar{x}_{[m],1}),\bar{\psi}_{\iota}=\langle\psi_{\iota,k}(\bar{x}_{[m],0},\dotsc,\bar{x}_{[m],k(\iota)}):k=1,\dotsc,k_{\iota}\rangle$
for $\iota<\iota_{*}$ in $\Delta$ such that for every $\lambda$

  1. $(a)_{\lambda}$

for every $\bar{f}=\langle
f_{\alpha}:\alpha<\lambda\rangle,f_{\alpha}:\alpha\rightarrow\iota_{*}$ we can
find $\bar{a}_{\alpha}\in{}^{m}{\mathfrak{C}}$ for $\alpha<\lambda$ such that

    1. $(\alpha)$

$\psi_{\iota,1}[\bar{a}_{\alpha},\bar{a}_{\beta}]$ for $\alpha<\beta<\lambda$
such that $f_{\beta}(\alpha)=\iota$

    2. $(\beta)$

$\psi_{\iota,k}(\bar{a}_{\alpha_{0}},\dotsc,\bar{a}_{\alpha_{k(\iota)-1}},\bar{a}_{\beta})$
when $\iota<\iota_{*},k=2,\dotsc,k_{\iota}$ and
$\alpha_{0}<\ldots<\alpha_{k(\iota)-1}<\beta<\lambda$ and
$f_{\beta}{\restriction}[\alpha_{0},\alpha_{k(\iota)-1}]$ is contantly $\iota$

  2. $(b)$

there are no $\bar{a}_{0},\dotsc,\bar{a}_{n}\in{}^{m}{\mathfrak{C}}$ such
that:

    1. $(\alpha)$

$\psi_{\iota,1}(\bar{a}_{\ell(1)},\bar{a}_{\ell(2)})$ when $\ell(1)<\ell(2)$

    2. $(\beta)$

$\psi_{\iota,k}(\bar{a}_{\ell(0)},\bar{a}_{\ell(1)},\dotsc,\bar{a}_{\ell(k-1)},\bar{a}_{n})$
when $\ell(1)<\ell(2)<\ldots<\ell(k(\iota)-1)<n$ and
$\iota=\eta(\ell(0))=\eta(\ell(1))$.

2) Relatives are as in Definitions 1.1, 1.3.

######  Remark 3.6.

To apply 3.5 we may replace $\mathbb{F}_{\lambda}$ by: for
$\mathbb{n}\leq\omega$

  1. $(*)$

$F^{\mathbb{n}}_{\lambda,\iota}$ is the set of $\bar{f}=\langle
f_{\alpha}:\alpha<\lambda\rangle$ such that
$f_{\alpha}:[\alpha]^{<\mathbb{n}}\rightarrow\iota$.

######  Remark 3.7.

Note that it does not matter if we use

  1. $(a)$

$T$ universal with JEP and amalgamation, $\Delta\subseteq{\rm qf}$, no
function symbols or

  2. $(b)$

$T$ compelte, $\Delta={\mathbb{L}}(\tau_{T})$.

Why?

1) Given a complete $T$, let $T^{\prime}$ be
$T\cup\\{(\forall\bar{x}_{[m]})(\varphi(\bar{x}_{[m]})\equiv
R_{\varphi(\bar{y}_{[m]})}(\bar{x}_{[m]})):\varphi(\bar{x})\in{\mathbb{L}}(\tau_{T})\\}$
where $\langle
R_{\varphi(\bar{x}_{[m]})}:\varphi(\bar{x}_{[m]})\in{\mathbb{L}}(\tau_{T})\rangle$
are new with no repetitions. Let $T^{\prime\prime}$ be the universal theory in
the vocabulary
$\tau^{\prime\prime}=\\{R_{\varphi(\bar{x}_{[m]})}:\varphi(\bar{x}_{[m]})\in{\mathbb{L}}(\tau_{T})\\}$
such that ${\rm Mod}_{T^{\prime\prime}}=\\{N^{\prime\prime}$: there is
$N^{\prime}\models T^{\prime}$ such that
$N^{\prime\prime}\subseteq(N^{\prime}{\restriction}\tau^{\prime\prime})\\}$.
So $T^{\prime}$ is complete with elimination of quantifiers and
$T^{\prime\prime}$ universal with amalgamation and JEP with no function
symbols and ${\rm univ}(\chi_{1},\lambda,T)={\rm
univ}(\chi_{1},\lambda,T^{\prime})={\rm
univ}(\chi_{1},\lambda,T^{\prime\prime})$, recalling the first is for $\prec$,
elementary embeddings and the second and third for $\subseteq$, embeddings.

2) If $T$ is universal (not complete) with the JEP (otherwise univerality is a
dull question) and amalgamation let $T^{\prime}={\rm Th}(M)$ for some
$M\in{\rm Mod}_{T}$ which is existentially closed.

Now

  1. $(a)$

${\rm univ}(\chi_{1},\lambda,T^{\prime})\leq{\rm univ}(\chi_{1},\lambda,T)$.

[Why? Let $\chi_{2}={\rm univ}(\chi_{1},\lambda,T)$ and $\langle
M_{\alpha}:\alpha<\chi_{2}\rangle$ exemplify it. For each $i$ there is
$N_{\alpha}$ such that $M_{\alpha}\subseteq N_{\alpha}\in{\rm Mod}_{T_{i}}$
and without loss of generality $\|N_{\alpha}\|=\lambda$, etc.]

  1. $(b)$

${\rm univ}(\chi_{1},\lambda,T)\leq{\rm univ}(\chi_{1},\lambda,T)$.

[Why? Also easy.]

## References

  * [LS77] Roger C. Lyndon and Paul E. Schupp, _Combinatorial group theory_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 89, Springer-Verlag, Berlin–Heidelberg–New York, 1977.
  * [Mir05] Džamonja Mirna, _Club guessing and the universal models_ , Notre Dame J. Formal Logic 46 (2005), 283–300.
  * [Sh:h] Saharon Shelah, _Classification Theory for Abstract Elementary Classes_ , Studies in Logic: Mathematical logic and foundations, vol. 18, College Publications, 2009.
  * [Sh:100] by same author, _Independence results_ , The Journal of Symbolic Logic 45 (1980), 563–573.
  * [GrSh:174] Rami Grossberg and Saharon Shelah, _On universal locally finite groups_ , Israel Journal of Mathematics 44 (1983), 289–302.
  * [KjSh:409] Menachem Kojman and Saharon Shelah, _Non-existence of Universal Orders in Many Cardinals_ , Journal of Symbolic Logic 57 (1992), 875–891, math.LO/9209201.
  * [Sh:420] Saharon Shelah, _Advances in Cardinal Arithmetic_ , Finite and Infinite Combinatorics in Sets and Logic, Kluwer Academic Publishers, 1993, N.W. Sauer et al (eds.). 0708.1979, pp. 355–383.
  * [Sh:457] by same author, _The Universality Spectrum: Consistency for more classes_ , Combinatorics, Paul Erdős is Eighty, vol. 1, Bolyai Society Mathematical Studies, 1993, Proceedings of the Meeting in honour of P.Erdős, Keszthely, Hungary 7.1993; A corrected version available as ftp: //ftp.math.ufl.edu/pub/settheory/shelah/457.tex. math.LO/9412229, pp. 403–420.
  * [Sh:500] by same author, _Toward classifying unstable theories_ , Annals of Pure and Applied Logic 80 (1996), 229–255, math.LO/9508205.
  * [Sh:589] by same author, _Applications of PCF theory_ , Journal of Symbolic Logic 65 (2000), 1624–1674, math.LO/9804155.
  * [DjSh:614] Mirna Dzamonja and Saharon Shelah, _On the existence of universal models_ , Archive for Mathematical Logic 43 (2004), 901–936, math.LO/9805149.
  * [Sh:702] Saharon Shelah, _On what I do not understand (and have something to say), model theory_ , Mathematica Japonica 51 (2000), 329–377, math.LO/9910158.
  * [ShUs:789] Saharon Shelah and Alex Usvyatsov, _Banach spaces and groups - order properties and universal models_ , Israel Journal of Mathematics 152 (2006), 245–270, math.LO/0303325.
  * [Sh:F1330] Saharon Shelah, _Universal models consistently exist: revisited_.
  * [Sh:F1414] by same author, _Minimal universality spectrum_.
  * [Sh:F1425] by same author, _Theorem with minimal universality spectrum_.