# Universality; new criterion for non-existence

Saharon Shelah  Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 9190401, 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: 2022-02-25)

###### Abstract.

We find new “reasons” for a class of models for not having a universal model
in a cardinal $\lambda$. This work, though has consequences in model theory,
is really in combinatorial (set theory). We concentrate on a prototypical
class which is a simply defined class of models, of combinatorial character -
models of $T_{{\rm ceq}}$ (essentially another representation of $T_{{\rm
feq}}$ which was already considered but the proof with $T_{{\rm ceq}}$ is more
transparent). Models of $T_{{\rm ceq}}$ consist essentially of an equivalence
relation on one set and a family of choice functions for it. This class is not
simple (in the model theoretic sense) but seems to be very low among the non-
simple (first order complete countable) ones. We give sufficient conditions
for the non-existence of a universal model for it in $\lambda$. This work may
be continued in [S+].

###### Key words and phrases:

classification theory; non-simple theories; universality; combinatorial set
theory

###### 2010 Mathematics Subject Classification:

Primary: 03E05, 03C45 Secondary: 03C50, 03C55

The author thanks Alice Leonhardt for typing an earlier version, and an
individual who wishes to remain anonymous for generously funding typing
services. We thank the ISF, The Israel Science Foundation grant 1838/19 for
partially supporting this research. This is paper number 1164 in the author’s
list. First typed/compiled December 12, 2018. References like [Sheb,
Th0.2=Ly5] means the label of Th.0.2 is y5.

Annotated Content

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

§1 For Mahlo cardinals, (label a), pg.1

§2 On Successor Cardinals and Club Guessing, (label b), pg.2

##  0\. Introduction

On a recent survey on the universality spectrum see [She21], an earlier survey
is [Dža05]; there have been several advances meanwhile (and this is one of the
advances after [She21]). See also [She20], noting the example there works also
for $\mu^{+}<2^{\mu}$ whenever $\mu$ is a strong limit singular. The problem
for general first order theories is a model theoretic one, but specific
examples are combinatorial set theoretic ones (and serve as proto-types for
suitable families of theories); so combinatorialists may ignore model
theoretic notions like “$T$ is simple, has the tree property, is ${\rm
TP}_{2}$”, and consider only the concrete universal theories considered; so
ignore 1.4 (1),(2) and their proof. Here we concentrate on the theory $T_{{\rm
ceq}}$, which we considered as a proto-typical “minimal” non-simple $T$, so
are expecting it (under $\leq_{{\rm univ}}$) to be low, so is it (like
$T_{{\rm feq}}$, see below), ${\rm NSOP}_{1}$, see [She93b], [DS04b], [SU08],
[CR16], [KR20], [KR18]). True, there were non-existence results near a strong
limit singular cardinal (see on the $T_{{\rm feq}}$ in [She93b], generalizing
it the oak property [DS06], [She17, §3]), but there were weak consistency
results on existence (see [She93b], [DS04a]). We had considered $T_{{\rm
feq}}$, a prototypical example of such theories, now $T_{{\rm ceq}}$ is
essentially equivalent to it for our aims, see 1.4(3),(4) but $T_{{\rm ceq}}$
seem more transparent; we intend to deal with “to what family of $T$’s
versions of our proof apply, in particular, ${\rm NTP}_{2}$ and non-simple”
elsewhere.

We have hoped/expected that for the $\lambda>\mu=\mu^{<\mu}$ but
$\lambda=\mu^{+}<2^{\mu}$ we shall have consistency results for theories like
$T_{{\rm feq}}$ and the class of triangle free graphs, [PS] and hopefull
[Shec].

We first give a case with stronger set theoretic assumptions, but more
transparent proof in §1. In §2 we give such proof under reasonable set
theoretic assumptions, (close to the so called club guessing) but then have to
consider finer points in combinatorial set theory on guessing clubs. Elsewhere
we hope to have relevant complimentary consistency (see [PS]) and families of
theories.

A priory we think that $T_{{\rm tfg}}$, the theory of triangle free graphs, is
“more complicated” then $T_{{\rm feq}},T_{{\rm ceq}}$, but now have doubts.

We thank Mark Poór and an anonymous referee for doing much to improve the
paper.

######  Question 0.1.

1) Does §1 apply to more theories than in §2?

2) Can we characterize the dividing line? Simple/non-simple in our context.

3) Does it help to have:

  1. $(*)$

for some $\mu,\mu<\lambda<2^{\mu}$ there is no
${\mathscr{A}}\subseteq[\lambda]^{\lambda}$ which is $\mu-{\rm AD}$ of
cardinality $>\lambda$?

This would justify the use of $\mu-{\rm AD}$ family
${\mathscr{A}}\subseteq[\lambda]^{\lambda}$ in some consistency results, see
[She90], [PS], see below.

######  Discussion 0.2.

Note that:

  1. $\boxplus$

if
$2<n\leq\omega,\theta\leq\mu\leq\lambda<2^{\theta},\lambda\nrightarrow[\mu]^{<n}_{\theta}$
and we let $T_{n}$ be the theory “$\\{P_{k}$ is an irreflexive asymmetric
$k$-place relation”: $k<n,k\geq 2\\}$ and $T_{n}$ has a universal model
$M_{*}$ in $\lambda$ then there is a $\mu$-disjoint
${\mathscr{A}}\subseteq[\lambda]^{\lambda}$ of cardinality $2^{\theta}$.

[Why? Without loss of generality the universe of $M_{*}$ is $\lambda.$ Let
${\mathbf{c}}:[\lambda]^{<n}\rightarrow\theta$ witness
$\lambda\nrightarrow[\mu]^{<n}_{\theta}$ and for $u\subseteq\theta$ let
$M_{u}=(\lambda,\dots P^{M_{u}}_{k},\ldots)_{k\in[2,n)}$ where
$P^{M_{u}}_{k}=\\{\eta\in{}^{k}\lambda:\eta$ is with no repetitions and
${\mathbf{c}}({\rm Rang}(\eta))\in u\\}$. So there is an embedding $f_{u}$ of
$M_{u}$ into $M_{*}$; now $\langle A_{u}=\\{{\rm
pr}(\alpha,f_{u}(\alpha):\alpha<\lambda\\}:u\subseteq\theta\rangle$ is a
family as promised when ${\rm pr}$ is a pairing function on $\lambda$. Why? If
$A_{u_{1}}\cap A_{u_{2}}$ has cardinality $\geq\mu$ and $u_{1}\not=u_{2}$ then
(letting $B=\\{\alpha<\lambda:f_{u_{1}}(\alpha)=f_{u_{2}}(\alpha)\\}$) without
loss of generality $u_{1}\nsubseteq u_{2}$ and $\mathbf{c}{\restriction}B$
omits any member of $u_{1}\setminus u_{2}$. The rest is left to the reader.]

###  0(A). Preliminaries

######  Notation 0.3.

1) $T$ is a theory with vocabulary $\tau_{T}=\tau(T)$ and is a first order, if
not said otherwise.

2)

  1. (a)

${\rm EC}_{T}=\\{M:M$ a model of $T\\}$,

  2. (b)

${\rm EC}_{T}(\lambda)=\\{M\in{\rm EC}_{T}:M$ of cardinality $\lambda\\}$,

  3. (c)

${\rm EC}_{T}(\lambda!)=\\{M\in{\rm EC}_{T}:M$ has universe $\lambda\\}$,

  4. (d)

for a set $A$ of ordinals and ordinal $\alpha$ let ${\rm suc}_{A}(\alpha)$ be
$\min\\{\beta\in A:\beta>\alpha\\}$.

3) Let ${\rm pr}$ be be an (easily computable) pairing function on ordinals
such that for $\alpha,\beta$ we have ${\rm
pr}(\alpha,\beta)<\max\\{\omega,\alpha+|\alpha|,\beta+|\beta|\\}$.

######  Convention 0.4.

1)

  1. (A)

If $T$ is a f.o. theory not complete (like $T^{0}_{{\rm ceq}},T^{0}_{{\rm
feq}},$ usually universal), then embedding are the usual ones, (on ${\rm
EC}_{T}$) and $\subseteq_{T}$ (on ${\rm EC}_{T}$) means $\subseteq$ and we
assume ${\rm EC}_{T}$ has amalgamation and ${\rm JEP}$.

  2. (B)

If $T$ is complete, then embeddings are elementary (on ${\rm EC}_{T}$) and
$\subseteq_{T}$ means $\prec$ on ${\rm EC}_{T}$.

  3. (C)

We say $f$ is a $T$-embedding of $M$ into $N$ or $f:M\longrightarrow_{T}N$
when $M,N$ are models of $T,f$ embed $M$ into $N$ and $f(M)\subseteq_{T}N$.

1A) In any case we always assume $T$ has ${\rm JEP}$ (for $\subseteq_{T}$ of
course).

2) If $\Delta\subseteq{\mathbb{L}}(\tau_{T})$ (such that $T$ has ${\rm JEP}$
under $\Delta-$embedding) then ${\rm univ}_{T,\Delta}(\lambda)$ is the minimal
$\chi$ such that there is a sequence $\bar{M}$ which is a
$(\lambda,T,\Delta)$-universal sequence which means:

  1. (a)

$\bar{M}=\langle M_{\alpha}:\alpha<\chi\rangle$ is a sequence of models of
$T$,

  2. (b)

each $M_{\alpha}$ is of cardinality $\lambda$,

  3. (c)

for every model $M$ of $T$ of cardinality $\lambda$ there is a
$\Delta$-embedding of $M$ into some $M_{\alpha}$, see below.

3) For given $T,\Delta$ as above and models $M,N$ of $T$, we say $f$ is a
$\Delta$-embedding of $M$ into $N$ when :

  1. (a)

$f$ is a function from $M$ into $N$,

  2. (b)

if $\varphi(x_{0},\dotsc,x_{n-1})\in\Delta$ and $a_{0},\dotsc,a_{n-1}\in M$
and $M\models\varphi[a_{0},\dotsc,a_{n-1}]$ then
$N\models\varphi[f(a_{0}),\dotsc,f(a_{n-1})]$,

  3. (c)

so $f$ is one-to-one when $(x\neq y)\in\Delta$.

4) For $T,\Delta$ as above in part (2) we may omit $\Delta$ when :

  1. (a)

$T$ is complete, $\Delta={\mathbb{L}}(\tau_{T})$, all first order formulas,

  2. (b)

$T$ not complete, $\Delta$ the set of quantifier free formulas in
${\mathbb{L}}(\tau_{T})$.

5) We may write ${\rm at},{\rm ep}$ for $\Delta_{{\rm
at}(T)}=\\{\varphi\in{\mathbb{L}}(\tau_{T}):\varphi$ is
atomic$\\},\Delta_{{\rm ep}(T)}=\\{\varphi\in{\mathbb{L}}(\tau_{T}):\varphi$
existential positive$\\}$ respectively. We may write $\tau$ instead of $T$. We
may write $\varphi$ instead $\Delta=\\{\varphi\\}$ and $\pm\varphi$ instead
$\Delta=\\{\varphi,\neg\varphi\\}$.

######  Notation 0.5.

1) Let $\alpha,\beta,\gamma,\delta,{\varepsilon},\zeta,\xi,i,j$ denote
ordinals.

2) Let $\kappa,\lambda,\mu,\chi,\partial,\theta,\Upsilon$ denote cardinals,
infinite if not said otherwise.

3) Let $k,\ell,m,n$ denote natural numbers.

4) Let $\varphi,\psi,\vartheta$ denote formulas, f.o. if not said otherwise.

5) For $\lambda>\kappa$ regular cardinals let
$S_{\kappa}^{\lambda}=\\{\delta<\lambda:{\rm cf}(\delta)={\rm cf}(\kappa)\\}$
and $S_{\leq\kappa}^{\lambda}=\\{\delta<\lambda:{\rm
cf}(\delta)\leq\kappa\\}.$

######  Definition 0.6.

1) $J^{{\rm bd}}_{\theta}=\\{A\subseteq\theta:\sup(A)<\theta\\}$, ${\rm bd}$
stands for bounding, for $\theta$ a regular cardinal or just a limit ordinal.

1A) For $\theta$ regular uncountable let:

  * •

$D^{{\rm club}}_{\theta}=\\{A\subseteq\theta$: there is a club (= closed
unbounded subset) $E$ of $\theta$ such that $E\subseteq A\\}$.

  * •

${\rm NS}_{\theta}$ is the non-stationary ideal on $\theta$.

2) For a regular $\theta$ let:

  1. (a)

${\mathfrak{d}\/}_{\theta}={\rm
Min}\\{|{\mathscr{F}}|:{\mathscr{F}}\subseteq{}^{\theta}\theta$ is $<_{J^{{\rm
bd}}_{\theta}}$-cofinal in $\mu_{\mu}\\}$

  2. (b)

${\mathfrak{b}}_{\theta}={\rm
Min}\\{|{\mathscr{F}}|:{\mathscr{F}}\subseteq{}^{\theta}\theta$ has no
$<_{J^{{\rm bd}}_{\theta}}$-upper bound$\\}$.

3) Let ${\mathfrak{d}\/}^{{\rm club}}_{\theta}$ be defined similarly using
$<_{{\rm NS}_{\theta}}$ when $\theta$ is regular uncountable.

4) For a model $M$ and a set $u\subseteq M$ let $M{\restriction}u$ is defined
naturally, allowing a function symbol to be interpreted as a partial function
(and so an individual constant to be not defined) but
$M{\restriction}u\subseteq M$ means $u={\rm c{\ell}}_{M}(u),$ see below.

5) For a model $M$ and $A\subseteq M$ let ${\rm c{\ell}}_{M}(A)={\rm
c{\ell}}(A,M)$ be the minimal subset $B$ of $M$ including $A$ and closed under
the functions of $M;$ so $M{\restriction}{\rm c{\ell}}_{M}(A)\subseteq M$ and
if $M$ has Skolem functions then $M{\restriction}{\rm c{\ell}}_{M}(A)\prec M.$

Recall

######  Definition 0.7.

1) For a regular uncountable cardinal $\lambda$ let
$\check{I}[\lambda]=\\{S\subseteq\lambda$: some pair $(E,\bar{a})$ witnesses
$S\in\check{I}(\lambda)$, see below$\\}$.

2) We say that $(E,u)$ is a witness for $S\in\check{I}[\lambda]$ if:

  1. $(a)$

$E$ is a club of the regular cardinal $\lambda$,

  2. $(b)$

$u=\langle u_{\alpha}:\alpha<\lambda\rangle,u_{\alpha}\subseteq\alpha$ and
$\beta\in u_{\alpha}\Rightarrow u_{\beta}=\beta\cap u_{\alpha}$,

  3. $(c)$

for every $\delta\in E\cap S,u_{\delta}$ is an unbounded subset of $\delta$ of
order-type $<\delta$ (and $\delta$ is a limit ordinal, necessarily $\delta$ is
not a regular cardinal).

3) For $\kappa=\mathrm{cf}(\kappa)<\lambda={\rm cf}(\lambda)$ let
$\check{I}_{\leq\kappa}[\lambda]$ be the ideal
$\\{S\subseteq\lambda:S\subseteq
S^{\lambda}_{\leq\kappa},S\in\check{I}[\lambda]\\}$

By ([She79], [She85] and better) [She93a] and [Shea] we have:

######  Claim 0.8.

Let $\lambda$ be regular uncountable.
1) If $S\in\check{I}[\lambda]$ then we can find a witness $(E,\bar{a})$ for
$S\in\check{I}[\lambda]$ such that (clauses (a), (b), (c) from 0.7(2) and):

  1. $(d)$

$\delta\in S\cap E\Rightarrow\,{\text{\rm otp\/}}(a_{\delta})=\,{\text{\rm
cf\/}}(\delta)$,

  2. $(e)$

if $\alpha\notin S$ then ${\text{\rm otp\/}}(a_{\alpha})<\,{\text{\rm
cf\/}}(\delta)$ for some $\delta\in S\cap E$.

2) $S\in\check{I}[\lambda]$ iff there is a pair $(E,\bar{\mathcal{P}})$ such
that:

  1. $(a)$

$E$ is a club of the regular uncountable $\lambda$,

  2. $(b)$

$\bar{\mathcal{P}}=\langle{\mathcal{P}}_{\alpha}:\alpha<\lambda\rangle$, where
${\mathcal{P}}_{\alpha}\subseteq\\{u:u\subseteq\alpha\\}$ has cardinality
$<\lambda$,

  3. $(c)$

if $\alpha<\beta<\lambda$ and $\alpha\in u\in{\mathcal{P}}_{\beta}$ then
$u\cap\alpha\in{\mathcal{P}}_{\alpha}$,

  4. $(d)$

if $\delta\in E\cap S$ then some $u\in{\mathcal{P}}_{\delta}$ is an unbounded
subset of $\delta$ of order type $<\delta$ (and $\delta$ is a limit ordinal).

3) We say a stationary subset $S$ has club guessing when some $\langle
C_{\delta}:\delta\in S\rangle$ witnesses it, which means: $C_{\delta}$ is a
club of $S$ and for every club $E$ of $\lambda$ for some $\delta\in S$ we have
$C_{\delta}\subseteq\lambda.$

##  1\. On $T_{{\rm ceq}}$ for Mahlo cardinals

As here we consider $T_{{\rm ceq}}$ the simplest, non-simple theory, we may
consider how much does it behave like the class of graphs (equivalently random
graph)? We prove that not by a non-existence result, but with quite specific
set theoretic assumptions.

$T_{{\rm ceq}}$ is very close to (and equivalent for our purposes to) the
older $T_{{\rm feq}}$ which is a prime example for a theory with the tree
order property, equivalently non-simple (even ${\rm TP}_{2}$ but having
neither the strict order property nor even just the ${\rm SOP}_{1}$). For it
we get here parallel and better results than [She93b] where it is proved that
there are limitations on the universality spectrum for $T_{{\rm feq}}$ and in
[DS06], which generalize the results for any $T$ with the so called oak
property, see somewhat more in [She17, §3]. The results in those papers are
meaningful when SCH fails, that is, consider a cardinal $\lambda$ such that:
for some strong limit singular $\mu,\mu^{+}<\lambda<2^{\mu}$ if $\lambda$ is
regular then “usually” $T_{{\rm feq}}$, has no universal in $\lambda$.

But what about $\lambda\in(\mu,2^{\mu})$ when for transparency we assume
$\mu=\mu^{<\mu}$?. Here (in §1) we get further such non-existence results for
(weakly inaccessible) Mahlo cardinals. In §2, we do better but the Mahlo case
may cover more classes, comes first and the proofs are more transparent. The
proof here (in §1) can be axiomatized as in §2 using:

  1. $\boxplus$

${\rm PGC}(\lambda,S)$ where $S$ is a stationary set of regular cardinals
$<\lambda$ means that some $\mathbf{U}$ witness ${\rm PGC}(\lambda,S)$ where
$\mathbf{U}=\\{\langle\omega(1+\varepsilon):\varepsilon<\theta\rangle:\theta\in
S\\}$ (so $\mathbf{U}=S$).

Recall that “$\mathbf{U}$ witness ${\rm PGC}(\lambda,\theta)$” means ${\rm
PGC}(\lambda,\theta)=\min\\{|\mathbf{U}|:\mathbf{U}\subseteq\mathbf{U}_{\lambda,\theta}\
\text{and}\ \mathbf{U}\\\ \text{does}\ $P-$\text{guess clubs}\\}.$ See 2.1(5),
Definition 2.1(3c) and 2.1(4).

First, recall (the reader can concentrate on the universal versions,
$T^{0}_{{\rm feq}},T^{0}_{{\rm ceq}}$, on $T_{{\rm feq}}$ see [She93b,
2.1=Lb3,3.1=Lc3]):

######  Definition 1.1.

$T_{{\rm feq}}=T^{1}_{{\rm feq}}$ is the model completion of the following
(universal first order) theory, $T^{0}_{{\rm feq}}$ which is defined by:

  1. (A)

$\tau=\tau(T^{0}_{{\rm feq}})$ consists of:

    1. (a)

predicates $P,Q$ (unary),

    2. (b)

$E$ (three place predicate written as $xE_{z}y$ instead $E(x,y,z)$),

  2. (B)

a $\tau$-model $M$ is a model of $T^{0}_{{\rm feq}}$ iff :

    1. (a)

the universe of $M$ is the disjoint union of $P^{M}$ and $Q^{M}$,

    2. (b)

$xE_{z}y\rightarrow P(z)\wedge Q^{M}(x)\wedge Q^{M}(y)$,

    3. (c)

for any fixed $z\in P^{M},E^{M}_{z}$ is an equivalence relation on $Q^{M}$.

######  Observation 1.2.

0) $T_{{\rm feq}}$ is well defined and ${\rm univ}(\lambda,T_{{\rm feq}})={\rm
univ}(\lambda,T^{0}_{{\rm feq}})$

1) So if $M\models T_{{\rm feq}}$ then :

  1. $(*)$

    1. (a)

in (B)(c) of Def. 1.1, for each $x\in P^{M},E_{x}^{M}$ is with infinitely many
equivalence classes,

    2. (b)

if $n<\omega,x_{1},\dotsc,x_{n}\in P^{M}$ with no repetition and
$y_{1},\dotsc,y_{n}\in Q^{M}$ then for some $y\in
Q^{M},\bigwedge\limits^{n}_{\ell=1}yE^{M}_{x_{\ell}}y_{\ell}$,

    3. (c)

if $n<\omega$ and $y_{1},\dotsc,y_{n}\in Q^{M}$ and $e$ is an equivalence
relation on $\\{1,\dotsc,n\\}$ then for some $x\in P^{M}$ we have
$y_{\ell}E^{M}_{x}y_{k}\Leftrightarrow\ell ek$,

    4. (d)

$P^{M},Q^{M}$ are infinite.

2) Hence $T_{{\rm feq}}$ has elimination of quantifiers and ${\rm
univ}_{T_{{\rm feq}}}(\lambda)={\rm univ}_{T^{0}_{{\rm feq}}}(\lambda)$.

We present a close relative, the main one we consider here (and, as proved
below, equivalent to $T_{{\rm feq}}$ for our purpose).

######  Definition 1.3.

$T_{{\rm ceq}}=T^{1}_{{\rm feq}}$ is the model completion of the following
(universal first order) theory, $T^{0}_{{\rm ceq}}$ which is defined by:

  1. (A)

$\tau=\tau(T^{0}_{{\rm ceq}})=\tau(T_{{\rm ceq}})$ consists of: $P,Q$ unary
predicates, $E$ a binary predicate and $F$ a binary function symbol,

  2. (B)

a $\tau$-model $M$ is a model of the universal theory. $T^{0}_{{\rm ceq}}$ iff
:

    1. (a)

$P^{M},Q^{M}$ is a partition of $M$,

    2. (b)

$E^{M}$ is an equivalence relation on $Q^{M}$,

    3. (c)

$F^{M}$ is a function from $Q^{M}\times P^{M}$ into $Q^{M}$ such that for
every $c\in P^{M},a\mapsto F^{M}(a,c)$ is choosing a representative for the
$a/E^{M}$-equivalence class, that is, we have:

      1. $(\alpha)$

$a\in Q^{M}\Rightarrow F^{M}(a,c)\in a/E^{M}$,

      2. $(\beta)$

if $a,b\in Q^{M}$ are $E^{M}$-equivalent then $F^{M}(a,c)=F^{M}(b,c)$.

      3. $(\gamma)$

if $c\notin P^{M}\vee a\notin Q^{M}$ then $F^{M}(a,c)$ is not defined (or, if
you prefer, is equal to $c$).

Concerning $\lambda$ in the neighborhood of a strong limit singular we shall
not give details as we can just quote.

######  Claim 1.4.

0) Concerning $T^{0}_{{\rm ceq}}$

  1. (a)

For a model $M$ of $T^{0}_{{\rm ceq}}$ and $A\subseteq M$ with $n$ elements,
the closure of $A$ inside $M$ has at most $n+n^{2}$ elements, (even at most
$n+(n/2)^{2}$ elements),

  2. (b)

$T^{0}_{{\rm ceq}}$ has amalgamation and JEP,

  3. (c)

$T^{0}_{{\rm ceq}}$ has a model completion, that is $T_{{\rm ceq}}$ is well
defined,

  4. (d)

${\rm univ}(\lambda,T_{{\rm ceq}})={\rm univ}(\lambda,T_{{\rm feq}})={\rm
univ}_{T^{0}_{{\rm ceq}}}(\lambda)$.

1) $T_{{\rm ceq}}$ is not simple, is ${\rm NSOP}_{2}$ and even ${\rm
NSOP}_{1}$ and has the oak property, in fact, by ${\rm qf}$ (quantifier free)
and even atomic formulas.

2) We have $(A)\Rightarrow(B)$ where :

  1. (A)

    1. (a)

$\theta<\mu<\lambda<\chi$,

    2. (b)

${\rm cf}(\lambda)=\lambda,\theta={\rm cf}(\theta)={\rm
cf}(\mu),\mu^{+}<\lambda$,

    3. (c)

$\chi:={\rm pp}_{\Gamma(\theta)}(\mu)>\lambda+|i^{*}|$,

    4. (d)

there is a set $\\{(a_{i},b_{i}):i<i^{*}\\}$ with
$a_{i}\in[\lambda]^{<\mu},b_{i}\in[\lambda]^{\theta}$ and
$|\\{b_{i}:i<i^{*}\\}|\leq\lambda$ such that: for every
$f:\lambda\rightarrow\lambda$ for some $i,f(b_{i})\subseteq a_{i}$,

  2. (B)

    1. (a)

$T_{{\rm ceq}}$ equivalently $T^{0}_{{\rm ceq}}$ has no universal model in
$\lambda$,

    2. (b)

Moreover, ${\rm univ}(\lambda,T_{{\rm ceq}})\geq\chi={\rm
pp}_{\Gamma(\theta)}(\mu)$.

3) $T_{{\rm feq}}$ can be interpreted in $T_{{\rm ceq}}$ hence ${\rm
univ}_{T_{{\rm feq}}}(\lambda)\leq{\rm univ}_{T_{{\rm ceq}}}(\lambda)$.

4) Also the inverse of part (3) holds.

###### Proof..

0) Easy as clause (d) follows by parts (3), (4).

1) By (1) (and part (3), (4)) quoting [DS06] where the oak property was
introduced.

2) Follows from parts (3), (4) and [She93b, Claim 2.2].

3) For a model $M$ of $T^{0}_{{\rm ceq}}$ we define a model $N=N[M]$ of
$T^{0}_{{\rm feq}}$ as follows:

  1. $(*)_{N,M}$

    1. (a)

$Q^{N}=P^{M},P^{N}=Q^{M}/E^{M}$,

    2. (b)

$E^{N}=\\{(a,b,C):C\in P^{N}=Q^{M}/E^{M}$ and $a,b\in Q^{N}$ and $(\forall
c\in C)[F^{M}(c,a)=F^{M}(c,b)]$ equivalently, $(\exists c\in
C)[F^{M}(c,a)=F^{M}(c,b)]\\}$.

Now check that $N\models T^{0}_{{\rm feq}}$ and $M\models
T_{ceq}\Leftrightarrow N\models T_{feq}$.

4) For a model $N$ of $T^{0}_{{\rm feq}}$ we define a model $M=M[N]$ of
$T^{0}_{{\rm ceq}}$ as follows:

  1. $(*)_{M,N}$

    1. (a)

$P^{M}=Q^{N}$ and $Q^{M}=\\{(c,A):c\in P^{N}$ and $A\in Q^{N}/E^{N}_{c}\\}$

    2. (b)

$E^{M}=\\{((c_{1},A_{1}),(c_{2},A_{2})):c_{1}=c_{2}\in P^{M}$ and
$A_{1},A_{2}\in Q^{N}/E^{N}_{c_{2}}\\}$

    3. (c)

$F^{M}:Q^{M}\times P^{M}\rightarrow Q^{M}$ is defined by: If $d\in Q^{M},b\in
P^{M}$ hence for some $c\in P^{N},A\in Q^{N}/E^{N}_{c}$ we have $d=(c,A)$ then
we let $F^{M}(d,b)=(c,b/E^{M}_{c}).$

Now check that $N\models T^{0}_{{\rm feq}}$ and $M\models T_{{\rm
ceq}}\Leftrightarrow N\models T_{{\rm feq}}$. ∎

We now point out a new reason involved “large ${\mathfrak{d}\/}_{\theta}$’s”
for not having a universal model in $\lambda$, even for many non-simple $T$’s.
In this section we deal with a case where the proof is simpler using $T_{{\rm
ceq}}$ and $\lambda$ a Mahlo cardinal.

######  Claim 1.5.

1) Assume $\lambda$ is a (weakly inaccessible) Mahlo cardinal and
$S=\\{\theta<\lambda:\theta$ regular (weakly inaccessible) and
${\mathfrak{d}\/}_{\theta}>\lambda\\}$ is stationary in $\lambda$ and $S$ has
club guessing.

Then

  1. (a)

${\rm univ}(\lambda,T_{{\rm ceq}})$ is $>\lambda$,

  2. (b)

even, $\geq\sup\\{\chi^{+}$: the set $\\{\theta\in
S:{\mathfrak{d}\/}_{\theta}>\chi\\}$ is stationary and has club guessing$\\}$.

2) We have $\chi<{\rm univ}(\lambda,T_{{\rm ceq}})$ when:

  1. (a)

$\lambda$ is a Mahlo weakly inaccessible cardinal,

  2. (b)

$\lambda\leq\chi$,

  3. (c)

$S\subseteq\\{\theta<\lambda:\theta$ is weakly inaccessible cardinal$\\}$ is
stationary.

  4. (d)

$\bar{{\mathscr{P}}}=\langle{\mathscr{P}}_{\theta}:\theta\in S\rangle$,

  5. (e)

if $\theta\in S$ then ${\mathscr{P}}_{\theta}$ is a set of $\leq\lambda$ clubs
of $\theta$,

  6. (f)

$\bar{{\mathscr{P}}}$ guess clubs of $\lambda$, that is, for every club $E$ of
$\lambda$ for some $C\in{\mathscr{P}}_{\theta},\theta\in S$ we have
$C\subseteq E$,

  7. (g)

$\mathfrak{d}_{\theta}>\chi$ for every $\theta\in S$.

###### Proof..

1) Clearly

  1. $(*)_{0}$

it suffices to:

    1. (a)

fix $\chi\geq\lambda$ such that $S_{\chi}=\\{\theta\in
S:\mathfrak{d}_{\theta}>\chi\\}$ is stationary and has club guessing,

    2. (b)

prove ${\rm univ}_{T}(\lambda)>\chi$.

Let $T=T_{{\rm ceq}},$ without less of generality assume $S=S_{\chi}$ and let:

  1. $(*)_{1}$

$\langle C_{\delta}:\delta\in S\rangle$ witness “$S$ has club-guessing”;

  2. $(*)_{2}$

if (A) below holds, then we define some objects in (B) where:

    1. (A)

      1. (a)

$M\in{\rm EC}_{T}(\lambda!)$,

      2. (b)

$|P^{M}|=\lambda$,

      3. (c)

$\theta$ is regular and $\theta\in S$,

      4. (d)

$E$ a club of $\theta$.

    2. (B)

we define:

      1. (a)

for $a\in P^{M}$ hence $a<\lambda$ let $g_{a}=g_{M,E,a}$ be the following
function from $\theta$ to $\theta$:

         * •

for $\alpha<\theta,g_{a}(\alpha)$ is the minimal $\beta\in E$ such that:
$\beta\in E\backslash(\alpha+1)$ and $(\beta_{1}\in
Q^{M}\cap\beta)\wedge(F^{M}(\beta_{1},a)<\theta)\Rightarrow
F^{M}(\beta_{1},a)<\beta$,

      2. (b)

${\mathscr{G}}^{0}_{M,E}=\\{g_{M,E,a}:a\in P^{M}\\}$, note that $E$ determine
$\theta$,

      3. (c)

for $\theta\in S$ let ${\mathscr{G}}^{*}_{M,\theta}=\\{g_{M,C_{\theta},a}:a\in
P^{M}\\}$.

Now easily

  1. $(*)_{3}$

for $a,M,\theta,E$ as above, $g_{M,E,a}$ is a well defined non-decreasing
function from $\theta$ into $\theta$, in fact, into $E\subseteq\theta,$

  2. $(*)_{4}$

if $M,N\in{\rm EC}_{T}(\lambda!)$ and $f$ embeds $M$ into $N$ then for some
club $E^{*}$ of $\lambda$: if $\theta\in
S,\theta=\sup(E^{*}\cap\theta),E\subseteq E^{*}\cap\theta$ is a club of
$\theta$ and $a\in P^{M}$ then $g_{M,E,a}\leq g_{N,E,f(a)}$ (so, the only way
$E^{*}$ influences is the demand “$E\subseteq E^{*}$”).

Recall that, $\theta\in S\Rightarrow{\mathfrak{d}\/}_{\theta}>\chi\geq\lambda$
and we shall prove that ${\rm univ}(\lambda,T)>\chi$; this suffices. So assume
$\langle M_{\alpha}:\alpha<\chi\rangle$ is a sequence of members of ${\rm
EC}_{T}(\lambda!)$.

So for each $\theta\in S$ the set
${\mathscr{G}}_{\theta}=\cup\\{{\mathscr{G}}^{*}_{M_{\alpha},\theta}:\alpha<\chi$
satisfies $|Q^{M_{\alpha}}\cap\theta|=\theta\\}$ has cardinality $\leq\chi$
recalling $\lambda\leq\chi$.

For $\theta\in S$, as $|{\mathscr{G}}_{\theta}|<{\mathfrak{d}\/}_{\theta}$,
necessarily there is an increasing $g_{\theta}\in{}^{\theta}\theta$ such that
$g\in{\mathscr{G}}_{\theta}\Rightarrow g_{\theta}\nleq g\mod J^{{\rm
bd}}_{\theta}$ and without loss of generality,
$g_{\theta}\in{{}^{\theta}(C_{\theta}}).$ Now we define a model $N\in{\rm
EC}_{T^{0}_{{\rm ceq}}}(\lambda!)$ with $\tau_{N}=\tau(T_{{\rm
ceq}})=\tau(T^{0}_{{\rm ceq}})$ as follows:

  1. (A)

universe is $\lambda$,

  2. (B)

    1. (a)

$Q^{N}$ is the set of odd ordinals $<\lambda$,

    2. (b)

$E^{N}$ is an equivalence relation on $Q^{N}$ such that for every
$\alpha<\beta<\lambda$ satisfying $\beta$ is divisible by $|\alpha|,\alpha\in
Q^{N}$ we have $|\alpha/E^{N}\cap\beta|=|\beta|$,

    3. (c)

if $\alpha=4\beta+1<\lambda$ then $\alpha/E^{N}$ is disjoint to $\alpha$,

    4. (d)

if $\theta\in S$ and $\alpha<\theta$ then
$\theta>F^{N}(4\alpha+1,\theta)>g_{\theta}(4\alpha+1)$.

This is easy to do.

To show that $\bar{M}$ does not witness ${\rm univ}(\lambda,T)\leq\chi$ is
suffice to show that $N$ cannot be embedded in $M_{\alpha}$ for any
$\alpha<\chi$. Toward contradiction assume that $\alpha<\chi$ and $f$ is an
embedding of $N$ into $M_{\alpha}$. Let $E=\\{\delta<\lambda:\delta$ a limit
ordinal such that
$((M_{\alpha}{\restriction}\delta,N{\restriction}\delta,f{\restriction}\delta,<{\restriction}\delta)\prec(M_{\alpha},N,f,<{\restriction}\lambda))\\}$.
Clearly $E$ is a club of $\lambda$ hence for some $\theta\in S_{\chi}$ we have
$C_{\theta}\subseteq E$. Let $h\in{}^{\theta}\theta$ be
$g_{M_{\alpha},C_{\theta},f(\theta)}$, so is well defined and belongs to
${\mathscr{G}}^{0}_{M_{\alpha},C_{\theta}}$ hence to
${\mathscr{G}}^{*}_{M_{\alpha},\theta}$ hence to ${\mathscr{G}}_{\theta}$
hence $g_{\theta}\not<h\mod J^{{\rm bd}}_{\theta}.$ Now,

  1. $\bullet_{1}$

choose $\alpha<\theta$ such that $h(\alpha)<g_{\theta}(\alpha),$

  2. $\bullet_{2}$

let $\gamma=4\alpha+1,$

  3. $\bullet_{3}$

$F^{N}(\gamma,\theta)\in(g_{\theta}(\alpha),\theta)$ by the choice of $N$ i.e.
(B)(d) above,

  4. $\bullet_{4}$

$g_{\theta}(\alpha)\in C_{\theta}$ by the choice of $g_{\theta},$

  5. $\bullet_{5}$

$C_{\theta}\subseteq E$ by the choice of $\theta,$

  6. $\bullet_{6}$

$g_{\theta}(\alpha)\in E$ by $\bullet_{4}$ and $\bullet_{5},$

  7. $\bullet_{7}$

every member of $E$ is closed under $f$ and $f^{-1},$

[Why? By the choice of $E$ it is closed under $f,$ as $f$ is one-to-one
similarly for $f^{-1}$.]

  1. $\bullet_{8}$

$f(F^{N}(\gamma,\theta))\in[g_{\theta}(\alpha),\theta),$

[Why? By $\bullet_{3},\bullet_{6}$ and $\bullet_{7}.$]

  1. $\bullet_{9}$

$f(F^{N}(\gamma,\theta))=F^{M_{\alpha}}(f(\gamma),f(\theta)),$

[Why? As $f$ embed $N$ into $M_{\alpha}.$]

  1. $\bullet_{10}$

$F^{M_{\alpha}}(f(\gamma),f(\theta))\in[g_{\theta}(\alpha),\theta),$

[Why? by $\bullet_{8}$ and $\bullet_{9}.$]

  1. $\bullet_{11}$

$h(\alpha)$ is a member of $C_{\theta},$ hence a limit ordinal and in $E,$

[Why? By the choice of $h.$]

  1. $\bullet_{12}$

$\alpha<h(\alpha),$ $\gamma<h(\alpha)$ and $f(\gamma)<h(\alpha),$

[Why? First, $\alpha<h(\alpha)$ by the choice of $h.$ Second,
$\gamma<h(\alpha)$: as $h(\alpha)$ is limit $>\alpha$ by $\bullet_{11}$ and
$\gamma=4\alpha+1$ by $\bullet_{1}.$ Third $f(\gamma)<h(\alpha)$: as
$\gamma<h(\alpha)$ and $h(\alpha)\in E$ by $\bullet_{11}$ and so $h(\alpha)$
is closed under $f$ by $\bullet_{7}.$]

  1. $\bullet_{13}$

$F^{M_{\alpha}}(f(\gamma),f(\theta))<h(\alpha),$

[Why? By the choice of $h$ as $g_{M_{\alpha},C_{\theta},f(\theta)}$ and
$\bullet_{12}$].

Now by the inequalities $\bullet_{1},\bullet_{8},\bullet_{9}$ and
$\bullet_{13}$ we get $h(\alpha)<g_{\theta}(\alpha)\leq
f(F^{N}(\gamma,\theta))=F^{M_{\alpha}}(f(\gamma),f(\theta))<h(\alpha),$
contradiction.

2) Similarly. ∎

######  Remark 1.6.

Under the assumption of 1.5(2), we can similarly prove that: for every
sequence $\langle(E_{\xi},{\mathscr{G}}_{\xi,\theta}):\xi<\chi,\theta\in
S\rangle$ satisfying clause (A) below, there is a sequence $\langle
g_{\theta,\alpha}:\theta\in S,\alpha<\lambda\rangle$ with
$g_{\theta,\alpha}\in{}^{\theta}\theta$ satisfying clause (B) below, where:

  1. (A)

$E_{\xi}$ is a club of $\lambda$ for $\xi<\chi$ and
${\mathscr{G}}_{\xi,\theta}\subseteq{}^{\theta}\theta$ has cardinality
$\leq\lambda$ for $\xi<\lambda,\theta\in S$.

  2. (B)

for every $\xi<\chi$ and club $E$ of $\lambda$ there are $\theta\in{\rm
acc}(E_{\xi})\cap E\cap S$ and $\alpha<\lambda$ such that
$\theta=\sup(E_{\xi}\cap E\cap\theta)$ and
$g\in{\mathscr{G}}_{\xi,\theta}\Rightarrow g_{\theta,\alpha}\nleq g\mod
J^{{\rm bd}}_{E_{\xi}\cap E\cap\theta}$.

##  2\. On successor Cardinals and club guessing

We first introduce the relevant notions (in 2.1); (we could add clause
2.1(2)(b) into the definition of ${\mathbf{U}}_{\lambda,\theta}$ in 2.1(1),
but so far it does not matter111Note that it is relevant to “fully $D$-guess
clubs” implies “almost guess clubs”, see 2.2). We then investigate it and use
it for sufficient conditions for “no universal”.

######  Definition 2.1.

Assume $\lambda>\theta$ are regular and $D\subseteq{\mathscr{P}}(\theta)$ is
upward closed non-empty satisfying $D\subseteq[\theta]^{\theta}$, omitting $D$
means $D=\\{\theta\\}$; and $\mathfrak{B}$ is a model with universe $\lambda$
and countable vocabulary but $\mathfrak{B}$ is locally finite when
$\theta={\aleph_{0}}$. Saying “for $D$-most $\varepsilon<\theta$” will mean
“for some $X\in D$ for every $\varepsilon\in X$”. The main case222 We may omit
clause (b) from the definition 2.1(3) of “fully $D$-guess clubs”, the only
problem this cause is for it implying the other versions, (see 2.2).  is
$\theta>{\aleph_{0}}$, this is necessary for the “full” cases (see parts (2)),
but not for the others; we may forget to assume $\theta>{\aleph_{0}}$.

1) Let ${\mathbf{U}}_{\lambda,\theta}=\\{\bar{u}:\bar{u}=\langle
u_{i}:i<\theta\rangle$ is $\subseteq$-increasing continuous, and
$i<\theta\Rightarrow i\subseteq u_{i}\in[\lambda]^{<\theta}$ (hence
$\theta\subseteq\cup\\{u_{i}:i<\theta\\}\in[\lambda]^{\theta})$ and
$\bigwedge\limits_{i<\theta}u_{i}\cap\theta\in\theta\\}.$

1A) We shall say that ${\mathbf{U}}\subseteq\mathbf{U}_{\lambda,\theta}$ obeys
$\mathfrak{B}$ when every $\bar{u}\in{\mathbf{U}}$ does, which means that for
every $\varepsilon<\theta$ we have $\mathfrak{B}\upharpoonright
u_{\varepsilon}\subseteq\mathfrak{B}$, (if $\mathfrak{B}$ has Skolem functions
this is equivalent to $\mathfrak{B}\upharpoonright
u_{\varepsilon}\prec\mathfrak{B}$ which implies $\theta>{\aleph_{0}}$).

2) We say ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ fully
$D$-guesses clubs when $\theta>{\aleph_{0}}$ and for every model $M$ with
universe $\lambda$ and countable vocabulary there is $\bar{u}\in{\mathbf{U}}$
which fully $D$-guesses $M$ meaning333 We may omit in 2.1(2) the clauses
$(a)(\alpha),(b)$ but then we have problems with ${``}{\rm FGC}\Rightarrow{\rm
AGC}$ and the gain is doubtful.

  1. (a)

    1. $(\alpha)$

if444This implies a case of club guessing. $\varepsilon<\theta$ then ${\rm
c{\ell}}(u_{\varepsilon},M)\subseteq\sup(u_{\varepsilon})$, moreover (actually
follows using an expansion of $M$) $M{\restriction}\sup(u_{\varepsilon})\prec
M,$

    2. $(\beta)$

$(\exists{\mathscr{X}}\in
D)(\forall{\varepsilon})[{\varepsilon}\in{\mathscr{X}}\Rightarrow{\rm
c{\ell}}_{M}(u_{\varepsilon})=u_{\varepsilon}\subseteq M]$, i.e. for $D$-most
$\varepsilon<\theta$ the set $u_{\varepsilon}$ is closed under the functions
of $M$, (in an equivalent definition
$M_{\varepsilon}{\restriction}u_{\varepsilon}\prec M$ as we can expand $M$ by
Skolem functions).

  2. (b)

the sequence ${\rm
ord}(\bar{u})=\langle\sup(u_{\varepsilon}):\varepsilon<\theta\rangle$ is
strictly increasing.

3) We say ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ almost
$D$-guesses clubs when :

  1. (a)

for every model $M$ with universe $\lambda$ and countable vocabulary and
$A\in[\lambda]^{\lambda}$ for some $\bar{u}\in{\mathbf{U}}$ we have:

    1. $(\alpha)$

if ${\varepsilon}<\theta$ then ${\rm
c{\ell}}(u_{{\varepsilon}},M)\subseteq\sup(u_{{\varepsilon}})$; as in
$(a)(\alpha)$ of part (2) without the moreover,

    2. $(\beta)$

for $D$-most $\varepsilon<\theta$ we have $A\cap
u_{{\varepsilon}+1}\nsubseteq\sup(u_{\varepsilon})$,

    3. $(\gamma)$

$c\ell(\bigcup\limits_{{\varepsilon}<\theta}u_{\varepsilon},M)=\bigcup\limits_{{\varepsilon}<\theta}u_{\varepsilon}$,
that is,
$M{\restriction}(\bigcup\limits_{{\varepsilon}<\theta}u_{\varepsilon})\subseteq
M$,

  2. (b)

if $\bar{u}\in{\mathbf{U}}$ then ${\rm
ord}(\bar{u})=\langle\sup(u_{\varepsilon}):{\varepsilon}<\theta\rangle$ is
strictly increasing.

3A) We say ${\mathbf{U}}$ medium $D$-guesses clubs when as in part (3)
omitting clause $(a)(\gamma)$.

3B) We say ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$
semi-$D$-guesses clubs when :

  1. (a)′

as (a) in part (3) but replacing $(\beta)$ by:

    1. $(\beta)^{\prime}$

for $D$-most ${\varepsilon}<\theta$ for some $\zeta\in[{\varepsilon},\theta)$
and $\alpha\in A$ we555 The “$\alpha\notin u_{\zeta}$” follows, and “$D$
-most” can be replaced by “all”.  have $\alpha\in(u_{\zeta+1}\backslash
u_{\zeta})\cap(\sup(u_{\varepsilon+1})\backslash\sup(u_{\varepsilon}))$,

  2. (b)

as in part (3).

3C) We say ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$
pseudo-$D$-guess clubs when :

  1. (a)′′

if $M$ is as above and $A\in[\lambda]^{\lambda}$ then for some
$\bar{u}\in{\mathbf{U}}$ we have:

    1. $(\alpha)$

as is part (3) clause $(a)(\alpha)$,

    2. $(\beta)$

for $D$-most ${\varepsilon}<\theta$ for some $\zeta\in[{\varepsilon},\theta)$
and $\alpha\in A$ we have $\alpha\in(u_{\zeta+1}\backslash
u_{\zeta})\cap(\sup(u_{\varepsilon+1})\backslash\sup(u_{\varepsilon}))$,

  2. (b)

as above.

3D) We say ${\mathbf{U}}$ is $(\lambda,\theta)$-reasonable (or just reasonable
when $(\lambda,\theta)$ are clear from the context) when
${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ satisfies clause (3)(b).

4) We say ${\mathbf{U}}$ does $X-D$-guess clubs when :

  * •

${\mathbf{U}}$ does fully $D$-guess clubs and $X=F$,

  * •

${\mathbf{U}}$ does almost $D$-guess clubs and $X=A$,

  * •

${\mathbf{U}}$ does semi-guess clubs and $X=S$,

  * •

${\mathbf{U}}$ does medium $D$-guess clubs and $X=M$,

  * •

${\mathbf{U}}$ does pseudo guess clubs and $X=P$.

5) Let ${\rm
XGC}_{D}(\lambda,\theta)=\min\\{|{\mathbf{U}}|:{\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$
and ${\mathbf{U}}$ does $X-D$-guess clubs$\\}$.

5A) Similarly ${\rm XGC}_{D}(\lambda,\theta,\mathfrak{B})$ when we restrict
ourselves to $\mathbf{U}$ obeying $\mathfrak{B}$.

6) We say ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ is bounded when
there is an $F$ witnessing it which means: $F$ is a function from
$\\{\bar{u}{\restriction}(\zeta+1):\bar{u}\in{\mathbf{U}},\zeta<\theta\\}$
into $\lambda$ such that
$F(\bar{u}_{1}{\restriction}(\zeta_{1}+1))=F(\bar{u}_{2}{\restriction}(\zeta_{2}+1))\Rightarrow\bar{u}_{1}{\restriction}(\zeta_{1}+1)=\bar{u}_{2}{\restriction}(\zeta_{2}+1)$
and $F(\bar{u}{\restriction}(\zeta+1))<\sup(u_{\zeta+1})$.

7) We say “strongly bounded” when in addition
$F(\bar{u}{\restriction}(\zeta+1))\in u_{\zeta+1}$ for every $\zeta<\theta$.

8) We say ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ is weakly
bounded, when there is a function $F$ witnessing it which means:

  1. (a)

${\rm Dom}(F)=\\{{\rm
ord}(\bar{u}{\restriction}(\zeta+1)):\bar{u}\in{\mathbf{U}}$ and
$\zeta<\theta\\}$ recalling ${\rm
ord}(\bar{u})=\langle\sup(u_{\varepsilon}):{\varepsilon}<\theta\rangle$,

  2. (b)

${\rm Rang}(F)\subseteq\lambda$ and $F({\rm
ord}(\bar{u}){\restriction}(\zeta+1))<\sup(u_{\zeta+1})$ for
$\bar{u}\in{\mathbf{U}}$ and $\zeta<\theta$,

  3. (c)

if $\zeta_{1},\zeta_{2}<\theta$ are successor of successor ordinals and
$\bar{u}_{1},\bar{u}_{2}\in{\mathbf{U}}$ and $F({\rm
ord}(\bar{u}_{1}){\restriction}\zeta_{1})=F({\rm
ord}(\bar{u}_{2}){\restriction}\zeta_{2})$ then ${\rm
ord}(\bar{u}_{1}){\restriction}\zeta_{1}={\rm
ord}(\bar{u}_{2}){\restriction}\zeta_{2}$.

9) Let

  1. (a)

if $\bar{u}\in{\mathbf{U}}_{\lambda,\theta}$ and $f:\theta\rightarrow\theta$
is $\leq$-increasing continuous with limit $\theta$ then
$\bar{u}^{[f]}=\bar{u}[f]:=\langle
u_{f({\varepsilon})}:{\varepsilon}<\theta\rangle]$,

  2. (b)

if ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ and
$f:\theta\rightarrow\theta$ is $\leq$-increasing continuous with limit
$\theta$ then
${\mathbf{U}}^{[f]}:={\mathbf{U}}[f]=\\{\bar{u}{[f]}:\bar{u}\in{\mathbf{U}}\\}$,

  3. (c)

if ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ and ${\mathscr{F}}$ is
a set of $\leq$-increasing continuous function from $\theta$ into $\theta$
with limit $\theta$ then
${\mathbf{U}}[{\mathscr{F}}]=\\{\bar{u}[f]:\bar{u}\in{\mathscr{F}},f\in{\mathscr{F}}\\}$,

  4. (d)

if $w\in[\theta]^{\theta}$ then $f_{w}=f[w]$ is the
$g:\theta\rightarrow\theta$ such that $\langle
g({\varepsilon}):{\varepsilon}<\theta$ but the closure of $w$ in order. so is
$\leq$-increasing continuous with limit $\theta$.

10) In (a),(b) of part (9) above we may write $\bar{u}[w],{\mathbf{U}}[w]$ for
$w\in[\theta]^{\theta}$ meaning $\bar{u}[f],{\mathbf{U}}[f]$ where $f=f_{w}$,
writing ${\mathbf{U}}[W],W\subseteq[\theta]^{\theta}$ mean
$\cup\\{{\mathbf{U}}[w]:w\in W\\}$.

11) Now for $X\in\\{F,A,S,M,P\\}$ we let (naturally and we can add
$\mathfrak{B}$ as in part (5A)):

  1. (a)

${\rm AXGC}_{D}(\lambda,\theta)={\rm
Min}\\{|{\mathbf{U}}|:{\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ does
$X-D$-guess clubs and is strongly bounded$\\}$,

  2. (b)

${\rm CXGC}_{D}(\lambda,\theta)$ is defined as in (a) but ${\mathbf{U}}$ is
just bounded,

  3. (c)

${\rm WXGC}_{D}(\lambda,\theta)$ is defined as in clause (a) but
${\mathbf{U}}$ is weakly bounded.

Some of the obvious implications are:

######  Observation 2.2.

1) If ${\mathbf{U}}$ fully $D$-guesses clubs, then ${\mathbf{U}}$ almost
$D$-guesses clubs,

2) If ${\mathbf{U}}$ almost $D$-guesses clubs then ${\mathbf{U}}$ semi-guess-
club and medium $D$-guesses clubs.

3) If ${\mathbf{U}}$ semi-$D$-guesses-clubs or medium $D$-guesses clubs then
${\mathbf{U}}$ does pseudo $D$-guesses clubs.

4) If $D_{1}\subseteq D_{2}\subseteq[\theta]^{\theta}$ then “$\mathbf{U}$ does
$X-D_{1}$-guess clubs” implies “${\mathbf{U}}$ does $X-D_{2}$-guess clubs” for
$X\in\\{F,A,M,S,P\\}$, we may write $\\{{\rm full},{\rm almost},{\rm
medium},{\rm semi},{\rm pseudo}\\}$.

5) Assume $\mathbf{U}\subseteq\mathbf{U}_{\lambda,\theta}$ and $\mathfrak{B}$
is as in 2.2. Then there is $\mathbf{U}^{\prime}$ such that:

  1. (a)

$\mathbf{U}^{\prime}\subseteq\mathbf{U}_{\lambda,\theta}$

  2. (b)

$|\mathbf{U}^{\prime}|\leq|\mathbf{U}|$

  3. (c)

if $\mathbf{U}$ does $X-D$-guess clubs, for $X\in\\{F,A,M,S,P\\}$ as in part
(4) then so does $\mathbf{U}^{\prime}$,

  4. (d)

$\mathbf{U}$ obeys $\mathfrak{B}$, (see 2.1(1)).

6) In 2.1(11) the number is $\geq\lambda$.

7) We may replace “countable vocabulary” by “vocabulary of cardinality
$<\theta$”.

###### Proof..

E.g.

5) Let $\mathbf{U}^{\prime}=\\{\bar{u}^{\prime}\in\mathbf{U}:$ for some
$\bar{u}\in\mathbf{U}$ for every $\varepsilon<\theta$ we have ${\rm
c{\ell}}_{\mathfrak{B}}(u_{\varepsilon})=u^{\prime}_{\varepsilon}\subseteq\sup(u_{\varepsilon})\\},$
it suffice to prove that $\mathbf{U}^{\prime}$ is as required. The main point
is to verify the appropriate version of clause (a) in Def 2.1. So let $M$ be a
model with universe $\lambda$ and countable vocabulary, we have to find a
suitable meber of $\mathbf{U}^{\prime}$. By renaming, without loss of
generality the vocabulary of $M$ is disjoint to the one of $\mathfrak{B}$ and
let $M^{\prime}$ be a common expansion of $M$ and $\mathfrak{B}$ with
$\tau(M^{\prime})=\tau(M)\cup\tau(\mathfrak{B})$. Let
$E=\\{\delta:M{\restriction}\delta\prec M\\}$. So
$(M^{\prime},E,<{\restriction}\lambda)$ is as required in clause (a) for
$\mathbf{U}$ hence there is a suitable $\bar{u}\in\mathbf{U}$. We can check
that in all cases $\bar{u}^{\prime}=\langle{\rm
c{\ell}}_{\mathfrak{B}}(u_{\varepsilon}):\varepsilon<\theta\rangle\in\mathbf{U}$
is as required here, so we are done.

7) Recall 2.1(1), the statement “$u_{\epsilon}\cap\theta\in\theta$”. ∎

######  Definition 2.3.

1) For the model theory: for a model $M\in{\rm
EC}_{T}(\lambda!),\Delta\subseteq{\mathbb{L}}(\tau_{T})$ and
$u\subseteq\lambda,A\subseteq M$ let $M^{[A]}{\restriction}_{\Delta}u$ be the
model $M{\restriction}u$ expanded by all the restriction to $u$ of all
relations definable by a $\Delta$-formula with parameters from $A$.

1A) If $\Delta={\mathbb{L}}_{{\rm qf}}(\tau_{M})$ then we may omit $\Delta$;
writing $\bar{a}$ instead $A$ means ${\rm Rang}(\bar{a})$.

2) For $M\in{\rm EC}_{T}(\lambda!),\bar{u}\in{\mathbf{U}}_{\lambda,\theta}$
and $\bar{a}\in{}^{\omega>}M$ let $g_{\bar{a},\bar{u},M}$ be the function from
$\theta$ to $\theta$ such that for $\zeta<\theta,g_{\bar{a},\bar{u},M}(\zeta)$
is the minimal ${\varepsilon}\in(\zeta,\theta)$ such that
$M^{[\bar{a}]}{\restriction}u_{\varepsilon}\prec
M^{[\bar{a}]}{\restriction}\bigcup\limits_{\xi<\theta}u_{\xi}$.

######  Claim 2.4.

We assume that $\mathfrak{B}$ is a model with universe $\lambda$ and countable
vocabulary, (for the case of full club guessing, we add locally finite when
$\theta={\aleph_{0}}.)$

1) We have

  1. (A)

${\rm CSGC}(\lambda,\theta)=\lambda$, moreover $\lambda={\rm
CSGC}(\lambda,\theta,\mathfrak{B})$ provided that:

     * •

$\lambda={\rm cf}(\lambda)=\theta^{++}$ and $\theta={\rm cf}(\theta)$

  2. (B)

${\rm ASGC}(\lambda,\theta)=\lambda$ provided666 We can weaken the demand: if
we weaken the demand in Definition 2.1(5) to “for stationary many
${\varepsilon}<\theta$” and $\theta\geq\aleph_{2}$.  that

     * •

$\lambda=\theta^{++},\theta={\rm cf}(\theta)$,

     * •

there is a stationary set $S\subseteq S^{\theta^{+}}_{\theta}$ from
$\check{I}_{\theta}[\theta^{+}]$,

  3. (C)

${\rm AFGC}(\lambda,\theta)=\lambda$ even with a reasonable witness. provided
that:

     * •

$\lambda=\lambda^{\theta}$ and $\theta={\rm cf}(\theta)>{\aleph_{0}}$,

  4. (D)

${\rm MGC}_{D}(\lambda,\theta)=\lambda$ when:

    1. (*)

$\theta={\rm cf}(\theta)<\lambda$ and there is ${\mathscr{S}}$ such that:

      1. (a)

${\mathscr{S}}\subseteq\\{w:w\subseteq\lambda,{\rm otp}(w)=\theta\\}$,

      2. (b)

${\mathscr{S}}$ has cardinality $\lambda$,

      3. (c)

if $A\in[\lambda]^{\lambda}$ then for some $w\in{\mathscr{S}}$ the set $w\cap
A$ has cardinality $\theta$,

      4. (d)

$D=[\theta]^{\theta}$.

  5. (E)

${\rm AGC}(\lambda,\theta)=\lambda$ when

  6. (*)

we have:

    1. (a),(b),(c),(d)

as in (D) above,

    2. (e)

the cofinality of $([\lambda]^{\theta},\subseteq)$ is equal to $\lambda$.

2) For regular $\lambda>\theta={\rm cf}(\theta)$ we have:

  1. (A)

if ${\rm SGC}_{D}(\lambda,\theta)=\lambda$ and
$\mathfrak{b}_{\theta}\leq\lambda$ then ${\rm
AGC}_{D}(\lambda,\theta)=\lambda$ when $D=[\theta]^{\theta}$,

  2. (B)

if ${\rm SGC}(\lambda,\theta))=\lambda$ and $\mathfrak{d}_{\theta}\leq\lambda$
then ${\rm AGC}(\lambda,\theta)=\lambda$ recalling that the default value of
$D$ is $\\{\theta\\}$.

3) For $\lambda>\theta={\rm cf}(\theta)$ such that777see footnote to part (2)
$\lambda>\theta^{+}$ we have ${\rm SGC}(\lambda,\theta)=\lambda$ provided that
(e.g. $\lambda=\theta^{+n}$ for some $n>0$):

  1. $\boxplus^{3}_{\lambda,\theta}$

${\rm cf}([\lambda]^{\theta},\subseteq)=\lambda$.

4) If ${\mathbf{U}}_{1}\subseteq{\mathbf{U}}_{\lambda,\theta}$ medium guesses
clubs, then there is ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$
which medium guesses clubs of cardinality $\leq|{\mathbf{U}}_{1}|$ and for
$\bar{u}\in\mathbf{U}$ we have:

  1. (a)

if $u=\cup\\{u_{i}:i<\theta\\}$ then $u\subseteq\delta=\sup(u)$ for some
$\delta<\lambda$ of cofinality $\theta$; (this actually follows by 2.1(3)(b)),

  2. (b)

if ${\mathfrak{b}}_{\theta}\leq\lambda$ then $u=\cup\\{u_{i}:i<\theta\\}$ and
$\bar{u}\in\mathbf{U}_{\lambda,\theta}$ then ${\rm
otp}(u\setminus\theta)=\theta$,

5) If $\lambda\geq\theta^{+}$ and $\theta={\rm cf}(\theta)>\aleph_{0}$ and
$S\subseteq\\{\delta<\lambda:{\rm cf}(\delta)=\theta\\}$ is stationary and
some $\bar{C}=\langle C_{\delta}:\delta\in S\rangle$ guesses clubs, then ${\rm
PGC}(\lambda,\theta)=\lambda$.

6) If ${\rm cf}([\lambda]^{\theta}.\subseteq)=\lambda,\theta>{\aleph_{0}}$ and
$\mathfrak{d}_{\theta}\leq\lambda$ then ${\rm FGC}(\lambda,\theta)=\lambda$,
moreover ${\rm BFGC}(\lambda,\theta)=\lambda$, (in fact, looking at [She93a]
we get strongly bounding).

######  Discussion 2.5.

1) In 2.4 we have ZFC results, we may get stronger results (on the full and
almost versions) in some forcing extensions see 2.14. and [S+].

2) We can look at the cases of Definition 2.1 for singular $\lambda$,
replacing $(u_{\zeta}\backslash\sup(u_{\varepsilon}))$ by $u_{\zeta}\backslash
u_{\varepsilon}$, but we have not arrive to it.

3) When we have clause (a)$(\gamma)$ of the Definition 2.1(3) there is less
need of clause $(a)(\alpha)$. E.g. in 2.4(1)(C) we do not need “$\lambda$
regular”.

4) In clauses (D), (E) of 2.1(1) we may add bounded/weakly bounded under
natural assumption.

###### Proof..

Without loss of generality $\mathfrak{B}$ has a pairing function ${\rm
pr}^{\mathfrak{B}}$ and its inverses as well as $\alpha+1,\alpha+\beta$ and
$\alpha\beta.$

1) Clause (A): First, choose $S,S^{+},\bar{C}$ such that (partial square
guessing clubs):

  1. $(*)_{1}$

    1. (a)

$S\subseteq\\{\delta<\lambda:{\rm cf}(\delta)=\theta$ and
$\delta>\theta^{+}\\}$ is stationary,

    2. (b)

$S\subseteq S^{+}\subseteq\\{\delta<\lambda:{\rm cf}(\delta)\leq\theta$ and
$\delta>\theta^{+}\\}$, moreover if $\delta\in S$ then
$\delta=\sup(S^{+}\cap\delta)$ ,

    3. (c)

$\bar{C}=\langle C_{\alpha}:\alpha\in S^{+}\rangle$,

    4. (d)

$C_{\alpha}$ is a closed subset of $\alpha$ of order type $\leq\theta$, and
${\rm otp}(C_{\alpha})$ is a limit ordinal iff $\alpha=\sup(C_{\alpha})$,

    5. (e)

for $\alpha\in S^{+}$ we have $\alpha\in S\Leftrightarrow{\rm
otp}(C_{\alpha})=\theta$,

    6. (f)

if $\alpha\in C_{\beta}$ then $\alpha\in S^{+}$ and
$C_{\alpha}=C_{\beta}\cap\alpha$,

    7. (g)

$\bar{C}\upharpoonright S$ guess clubs, i.e.: if $E$ is a club of $\lambda$
then for stationarily many $\delta\in S$ we have $C_{\delta}\subseteq E$,

    8. (h)

if $\alpha\in S^{+}$ then $\alpha>\theta^{+}$ and $\alpha$ is closed under
$\mathfrak{B}$, that is
$\mathfrak{B}{\restriction}\alpha\subseteq\mathfrak{B}$,

    9. (i)

if $\alpha\in S^{+}$ then $\theta^{2}$ divides $\delta$.

Why do they exist (provably in ${\rm ZFC}$)? see [Shea, 1.3=L1.3(b)], but we
elaborate (for the case $\theta>{\aleph_{0}}$); by [She91, 4.4(1), pg.47]
(with $\theta^{+},\lambda$ here standing for $\lambda,\lambda^{+}$ there):

  1. $(*)_{1.1}$

there are $W,\overline{S},\overline{C}_{i}\ (i<\theta^{+})$ such that:

    1. (A)

$W=\\{\delta<(\theta^{+})^{+}:{\rm cf}(\delta)<\theta^{+}\\}$ hence is in
$\check{I}[(\theta^{+})^{+}].$

    2. (B)

$W$ is the union of $\lambda$ sets which have the square property, i.e., there
are sequences $\overline{S}=\langle S_{i}:i<\lambda\rangle$ and
$\overline{C}_{i}=\langle C_{\delta}^{i}:\delta\in S_{i}\rangle$ for
$i<\lambda$ such that:

      1. (a)

$W\subseteq\bigcup_{i<\lambda}S_{i},$

      2. (b)

For $\delta\in S_{i},$ $C_{\delta}^{i}$ is a subset of $\delta\cap S_{i}$ of
cardinality $<\lambda$ closed in $\delta,$ and if $\delta$ is a limit ordinal
then $C_{\delta}^{i}$ is unbounded in $\delta,$

      3. (c)

For all $\delta_{1},\delta_{2}$ if $\delta_{2}\in S_{i}$ and $\delta_{1}\in
C_{\delta_{2}}^{i}$ then $\delta_{1}\in S_{i}$ and
$C_{\delta_{1}}^{i}=C_{\delta_{2}}^{i}\cap\delta_{1}.$ (Notice that
$\delta_{1}$ may also be a successor ordinal.)

Easily (and as in [Shear, a. III] making $\theta^{+}$ tries):

  1. $(*)_{1.2}$

there are $\zeta<\theta^{+},\ i<\lambda$ and a club $E_{*}$ of $\lambda$ such
that: for every club $E\subseteq E_{*}$ of $\lambda$ for some $\delta$ we have
$\delta\in S_{i},\ {\rm cf}(\delta)=\theta,\ \delta=\sup(C_{\delta}^{i}\cap
E_{*}),\ C_{\zeta}^{i}\cap E=C_{\delta}\cap E_{*}$ and ${\rm
otp}(C_{\delta}^{i}\cap E_{*})=\zeta.$

  2. $(*)_{1.3}$

without loss of generality $\alpha\in E_{*}\Rightarrow({\rm
c{\ell}}_{\mathfrak{B}}(\alpha)=\alpha)\wedge(\theta^{+})^{2}\mid\alpha\wedge\alpha\geq\theta^{+},$

  3. $(*)_{1.4}$

let:

    1. (a)

$e\subseteq\zeta$ is unbounded in $\zeta$ and ${\rm otp}(e)=\theta$

    2. (b)

$S=\\{\delta\in S_{i}:{\rm cf}(\delta)=\theta,\ {\rm otp}(C_{\delta}^{i}\cap
E_{*})=\zeta\\},$

    3. (c)

$S^{+}=\\{\alpha:\alpha\in S\ \text{or for some}\ \delta\in S\ \text{we have}\
\alpha\in C_{\delta}^{i}\ \text{and}\ {\rm otp}(\alpha\cap C_{\delta}^{i})\in
e\\}.$

    4. (d)

$C_{\delta}=\\{\alpha\in C_{\delta}^{i}:{\rm otp}(\alpha\cap
C_{\delta}^{i})\in e\\}$ for $\delta\in S.$

Now $S,\delta^{+},\langle C_{\delta}:\delta\in S^{+}\rangle$ satisfies all the
demands, proving $(*)_{1}.$

  1. $(*)_{2}$

For $\delta\in S$ let
$\langle\gamma^{\bullet}_{\delta,{\varepsilon}}:{\varepsilon}<\theta\rangle$
list $C_{\delta}$ in increasing order.

Second, fix $\bar{f},\bar{g}$ such that:

  1. $(*)_{3}$

    1. (a)

$\bar{f}=\langle f_{\alpha}:\alpha\in[\theta^{+},\lambda)\rangle$,

    2. (b)

$f_{\alpha}$ is a one-to-one function from $\theta^{+}$ onto $\alpha$,

    3. (c)

$\bar{g}=\langle g_{\xi}:\xi\in[\theta,\theta^{+})\rangle$,

    4. (d)

$g_{\xi}$ is a one-to-one function from $\theta$ onto $\xi$.

Third,

  1. $(*)_{4}$

    1. (a)

for $\delta\in S$ let $e_{\delta}=\\{\xi<\theta^{+}$: if $\alpha\in
C_{\delta}$ then ${\rm Rang}(f_{\alpha}{\restriction}\xi)=\alpha\cap{\rm
Rang}(f_{\delta}{\restriction}\xi)$ and this set includes
$C_{\delta}\cap\alpha$ and has cardinality $\theta\\}$

    2. (b)

$e_{\delta}$ is a club of $\theta^{+}$.

[Why clause (b) holds? As ${\rm otp}(C_{\delta})=\theta$ and $\alpha\in
C_{\delta}\cup\\{\delta\\}\Rightarrow|\alpha|=\theta^{+}$, this should be
clear.]

  1. $(*)_{5}$

for $\delta\in S$ and $\xi\in e_{\delta}$ let:

    1. (a)

$u_{\delta,\xi}={\rm Rang}(f_{\delta}{\restriction}\xi)$, it belongs to
$[\delta]^{\theta}$ and it includes $C_{\delta}$,

    2. (b)

we choose $\bar{u}_{\delta,\xi}=\langle
u_{\delta,\xi,{\varepsilon}}:{\varepsilon}<\theta\rangle$ by
$u_{\delta,\xi,{\varepsilon}}=c{\ell}_{\mathfrak{B}}(\\{f_{\gamma^{\bullet}_{\delta,\upsilon}}(g_{\xi}(\zeta)):\upsilon<\omega(1+{\varepsilon})$
and
$\zeta<\omega(1+\varepsilon)\\}\cup\\{\gamma^{\bullet}_{\delta,\upsilon}:\upsilon<\omega(1+\varepsilon)\\})$,

    3. (c)

for $w\in[\theta]^{\theta}$ let $\bar{u}^{[w]}_{\delta,\xi}$ be $\langle
u^{[w]}_{\delta,\varepsilon};\varepsilon<\theta\rangle$ where
$u^{[w]}_{\delta,\varepsilon}=u_{\delta,\iota}$ where: $\iota\in w$ is the
minimal $\iota$ that satisfies ${\rm otp}(w\cap\iota)=\varepsilon$, this fits
2.1(9)(d).

Note that (recalling $(*)_{2}$):

  1. $(*)_{6}$

For $\delta\in S,\xi\in e_{\delta}$ we have:

    1. (a)

$\bar{u}_{\delta,\xi}$ is a $\subseteq$-increasing continuous sequence of
subsets of $u_{\delta,{\varepsilon}}$,

    2. (b)

each $u_{\delta,\xi,{\varepsilon}}$ include
$C_{\gamma^{\bullet}_{\delta,\omega(1+\varepsilon)}}$ and is an unbounded
subset of $\gamma^{\bullet}_{\delta,\omega(1+\varepsilon)}$ and it is of
cardinality $<\theta$,

    3. (c)

$\cup\\{u_{\delta,\xi,{\varepsilon}}:{\varepsilon}<\theta\\}$ is equal to
$u_{\delta,\xi},$

    4. (d)

$u_{\delta,\xi,{\varepsilon}}$ is computable from ${\rm
pr}^{\mathfrak{B}}(\gamma^{\bullet}_{\delta,{\varepsilon}},\xi)$ recalling
that ${\rm pr}^{\mathfrak{B}}$ is a pairing function, using as parameters
$\bar{f},\bar{g}$ which were fixed in $(*)_{2}$.

[Why? should be clear.]

Lastly,

  1. $(*)_{7}$

let:

    1. (a)

$\mathbf{U}=\\{\bar{u}_{\delta,\xi}:\delta\in S,\xi\in C_{\delta}\\}$

    2. (b)

${\mathbf{U}}_{w}=\\{\bar{u}^{[w]}_{\delta,\xi}:\delta\in S$ and $\xi\in
e_{\delta}\\}$ for $w\in[\theta]^{\theta}$.

We shall prove that (why the $w$? for the use in the proof of part (4) of the
claim):

  1. $(*)_{8}$

if $w\in[\theta]^{\theta}$ then ${\mathbf{U}}_{w}$ witnesses ${\rm
WSGC}(\lambda,\theta)\leq\lambda$.

Fix $w$ now and we shall deal with all the demands:

  1. $(*)_{8.1}$

${\mathbf{U}}_{w}$ has cardinality $\leq\lambda$; in fact is equal to
$\lambda$.

[Why? As $|{\mathbf{U}}_{w}|\leq|\\{(\delta,\xi):\delta\in S,\xi\in
e_{\delta}\subseteq\theta^{+}\\}|\leq\lambda+\theta^{+}=\lambda$. The other
inequality is also easy as $\cup\\{u_{\delta,\xi}:\delta\in S,\xi\in
e_{\delta}\\}=\lambda$ and each $u_{\delta,\xi}$ has cardinality
$\theta<\lambda$.]

  1. $(*)_{8.2}$

${\mathbf{U}}_{w}\subseteq{\mathbf{U}}_{\lambda,\theta}$ is reasonable.

[Why? By the choices above.]

  1. $(*)_{8.3}$

${\mathbf{U}}_{w}$ semi-guess clubs.

[Why? Let $M$ and $A$ be as in Definition 2.1(3B)(a)′; without loss of
generality $M$ expand $\mathfrak{B}$ and let $M^{+}$ be the expansion of $M$
by the relation $<^{M^{+}}$, the order of the ordinals $<\lambda$ and
$P^{M^{+}}=A$, and let $E:=\\{\delta<\lambda:M^{+}{\restriction}\delta\prec
M^{+}\\}$, clearly $E$ is a club of $\lambda$. By the choice of $\bar{C}$
there is $\delta\in S$ such that $C_{\delta}\subseteq E$ (hence $\delta\in
E$). Note that if $\alpha\in C_{\delta}$ then $A\cap\alpha$ is unbounded in
$\alpha$.

Now recall that $M{\restriction}\delta\prec M$, $\langle u_{\delta,\xi}:\xi\in
e_{\delta}\rangle$ is $\subseteq$-increasing continuous with union $\delta$,
each $u_{\delta,\xi}$ is of cardinality $\leq\theta$ and $e_{\delta}$ is a
club of $\theta^{+}$ hence $e=\\{\xi\in
e_{\delta}:M^{+}{\restriction}u_{\delta,\xi}\prec M^{+}\\}$ is a club of
$\theta^{+}$. So if $\xi\in e$ then $A\cap u_{\delta,\xi}$ is unbounded in
$u_{\delta,\xi}$. Now choose $\xi\in e$, so $\bar{u}=\bar{u}_{\delta,\xi}$ is
as required.]

  1. $(*)_{8.4}$

${\mathbf{U}}$ is weakly bounded.

[Why? Just think, recalling $(*)_{1}$ and Definition 2.1(8), that is, note
that $\langle C_{\delta}\cap\alpha:\delta\in S^{+}\rangle$ has cardinality
$\leq\theta^{+}$ for each $\alpha<\lambda$ because $\beta\in
C_{\delta_{1}}\cap C_{\delta_{2}}\Rightarrow
C_{\delta_{1}}\cap\beta=C_{\delta_{2}}\cap\beta$ and
$\delta>\alpha\Rightarrow\sup(C_{\delta}\cap\alpha)\in C_{\delta},$ anyhow
below we shall get more.]

  1. $(*)_{9}$

${\mathbf{U}}$ is bounded hence ${\rm CSGC}(\lambda,\theta)$ holds, in fact:

    1. (a)

if
$u_{1}=u_{\delta_{1},\xi_{1},{\varepsilon}_{1}},u_{2}=u_{\delta_{2},\xi_{2},{\varepsilon}_{2}}$
and ${\rm pr}(\gamma^{\bullet}_{\delta_{1},{\varepsilon}_{1}},\xi_{1})={\rm
pr}(\gamma^{\bullet}_{\delta_{2},{\varepsilon}_{2}},\xi_{2})$ then:

      1. $(\alpha)$

$\langle\gamma^{\bullet}_{\delta_{1},{\varepsilon}}:{\varepsilon}\leq{\varepsilon}_{1}\rangle=\langle\gamma^{\bullet}_{\delta_{2},{\varepsilon}}:{\varepsilon}\leq{\varepsilon}_{2}\rangle$

      2. $(\beta)$

$u_{1}=u_{2}$

    2. (b)

${\rm
pr}(\gamma^{\bullet}_{\delta,{\varepsilon}},\xi_{1})<\gamma^{\bullet}_{\delta,{\varepsilon}+1}$.

[Why? Clause (a) holds by $(*)_{6}(d)$ and clause (b) by $(*)_{1}(h)$.]

We have finished proving $\lambda={\rm CSGC}(\lambda,\theta)$, and even ${\rm
CSGC}(\lambda,\theta,\mathfrak{B})$, that is clause (A) of part (1).

Clause (B): Fix a stationary $S\subseteq S^{\theta^{+}}_{\theta}$ which
belongs to $\check{I}_{\theta}[\theta^{+}]$, see Def 0.7. By 0.8 we can choose
$\langle\zeta_{\xi,{\varepsilon}}:\varepsilon<\theta\rangle$ for $\xi\in S$
such that for any such $\xi\in
S,\langle\zeta_{\xi,{\varepsilon}}:{\varepsilon}<\theta\rangle$ is increasing
continuous with limit $\xi$ and
$(\zeta_{\xi_{1},\varepsilon+1}=\zeta_{\xi_{2},\varepsilon+1})\wedge(\upsilon\leq\varepsilon)\Rightarrow\zeta_{\xi_{1},\upsilon}=\zeta_{\xi_{2},\upsilon}$
(and more). Without loss of generality this is $\overline{C}=\langle
C_{\alpha}:\alpha<\lambda$ such that if $\xi\in S,\ {\varepsilon}<\theta$ then
for some $\alpha<\zeta_{\xi,{\varepsilon}}$ we have
$\\{\zeta_{\xi,i}:i\leq{\varepsilon}\\}=C_{\alpha}$ and for
$\overline{u}\in{\mathbf{U}}$ such that
$\zeta=\sup(\bigcup_{{\varepsilon}}u_{{\varepsilon}})$ we require that
$\alpha\in u_{{\varepsilon}+1}$ (and $\overline{C}$ definable in
$\mathfrak{B}$).

Now in the proof of clause (A) of part (1) we choose $\bar{f},\bar{g}$ as in
$(*)_{3}$ but in addition $\bar{g}=\langle
g_{\alpha}:\alpha\in[\theta,\theta^{+}\rangle)$ satisfies that: if
$\alpha=\zeta_{\xi,\varepsilon},\varepsilon$ a limit ordinal then $g_{\alpha}$
is computable from $\langle
g_{\beta}:\beta\in\\{\zeta_{\xi,\iota}:\iota<\varepsilon\\}\rangle$ e.g. as
follows: for $\iota<\theta$ let $W_{\alpha}^{\iota}=\\{\gamma<\alpha:\
\text{for some}\ \beta\in\\{\zeta_{\xi,e}:e<\varepsilon\\}\ \text{we have}\
g_{\beta}(\gamma)<\iota\\}$ and then define $g_{\alpha}^{\iota}$ by induction
on $\iota<\theta$ such that:

  1. $(*)$

    1. (a)

$g_{\alpha}^{\iota}$ is a function from $W_{\alpha}^{\iota}$ onto some ordinal
$<\theta.$

    2. (b)

$g_{\alpha}^{\iota}$ is increasing with $\iota.$

    3. (c)

$g_{\alpha}^{\iota}{\restriction}(W_{\alpha}^{\iota}\setminus\bigcup\\{W_{\alpha}^{j}:j<\iota\\})$
is order preserving.

Also we can restrict ourselves to $\xi\in S$ such that $u_{\delta,\xi}$ is
closed under ${\rm pr}$. Then we can restrict ourselves to $(w,\delta,\xi)$
such that ${\varepsilon}_{1}<{\varepsilon}_{2}\in w\Rightarrow{\rm
pr}(\gamma^{\bullet}_{\delta,{\varepsilon}_{1}},\zeta_{\xi,{\varepsilon}_{1}})\in
u_{\delta,\xi,{\varepsilon}_{2}}$.

Clause (C): Easy but we elaborate.

We are assuming $\lambda=\lambda^{\theta},\theta={\rm cf}(\theta)$; so
$\mathbf{U}=\mathbf{U}_{\lambda,\theta}$ is trivially a subset of
$\mathbf{U}_{\lambda,\theta}$ of cardinality $\lambda$ and let $F$ be a one-
to-one function from
$\\{\overline{u}{\restriction}\epsilon:\overline{u}\in\mathbf{U}\ \text{and}\
\epsilon<\theta\\},$ clearly exist. Let $M$ be a model with universe $\lambda$
and we have to find $\bar{u}$ as promised. Toward this we choose
$u_{\varepsilon}$ by induction on $\varepsilon$ as follow:

  1. (a)

$u_{\varepsilon}$ is a subset of $\lambda$ of cardinality $<\theta$,

  2. (b)

$u_{\varepsilon}={\rm c{\ell}}(u_{\varepsilon},M)$ and has no last member,

  3. (c)

if $\varepsilon=\zeta+1$ then some $\alpha\in A\setminus\sup(u_{\zeta})$
belongs to $u_{\varepsilon}$,

  4. (d)

if $\varepsilon$ is a limit ordinal then
$u_{\varepsilon}=\cup\\{u_{\zeta}:\zeta<\varepsilon\\}$

  5. (e)

If $\epsilon=\zeta+1$ then $F(\overline{u}{\restriction}\epsilon)\in
u_{\epsilon}.$

There is no problem to carry the induction and $\langle
u_{\varepsilon}:\varepsilon<\theta\rangle$ is as required.

Clause (D) Recall that ${\mathscr{S}}\subseteq\\{w\subseteq\lambda:{\rm
otp}(w)=\theta\\}$ and more by our assumption. For each $w\in{\mathscr{S}}$
let $\bar{u}_{w}$ be $\langle u_{w,\varepsilon}:\varepsilon<\theta\rangle$
were $u_{w,\varepsilon}=\\{\alpha\in W:{\rm
otp}(w\cap\alpha)<\omega(1+\varepsilon)\\}\cup\varepsilon$. Now let
$\mathbf{U}=\\{\bar{u}_{w}:w\in{\mathscr{S}}\\}$, it suffice to prove that
$\mathbf{U}$ witness ${\rm MGC}(\lambda,\theta)=\lambda$.

Clearly most demands hold: $\mathbf{U}$ is a subset of
$\mathbf{U}_{\lambda,\theta}$ of cardinality $\lambda$, and for each
$\bar{u}\in\mathbf{U}$ the sequence $\langle
sup(u_{\varepsilon}):\varepsilon<\theta\rangle$ is increasing. The main point
is, to be given $M,A$ as in clause (a) of Def 2.1(3) and to prove that sub-
clauses $(\alpha),(\beta)$ there hold.

Let $M^{+}$ be an expansion of $M$ by the order $<^{M^{+}}$ of the ordinals
$<\lambda$, $R^{M^{+}}=A,{\rm pr}$ and let
$E=\\{\delta<\lambda:M^{+}{\restriction}\delta\prec M^{+}\\}$, clearly it is a
club of $\lambda$. Now there is no harm in replacing $A$ by a smaller sub-set
so let $A^{\prime}=\\{\alpha\in A:\alpha=\min(A\setminus\beta)$ for some
$\beta\in E\\}$. Clearly $A^{\prime}\in[\lambda]^{\lambda}$ so by the choice
of ${\mathscr{S}}$ there is $w\in{\mathscr{S}}$ such that $w\cap A$ has
cardinality $\theta$.

Now $\bar{u}_{w}\in\mathbf{U}$ is as required.

Clause E:

By [She93a] there is a stationary ${\mathscr{A}}\subseteq[\lambda]^{\theta}$
of cardinality $\lambda$, see details in the proof of part (3). Now for each
$w\in{\mathscr{S}}$ let $\mathscr{A}_{w}=\\{v\in{\mathscr{A}}:w\subseteq
v\\}$, so it is non-empty. Now for each $w\in{\mathscr{S}}$ let
$\langle\alpha_{w,\varepsilon}:\varepsilon<\theta\rangle$ list the members of
$w$ in increasing order. Also for each such pair $(w,v)$ let
$\bar{u}_{w,v}=\langle u_{w,v,\varepsilon}:\varepsilon<\theta\rangle$ be such
that:

  1. (a)

$u_{w,v,\varepsilon}$ is a subset of $v$ of cardinality $<\theta$,

  2. (b)

$u_{w,v,\varepsilon}$ is increasing continuous with $\varepsilon$,

  3. (c)

$u_{w,v,\varepsilon}$ includes
$\\{\alpha_{w,\zeta}:\zeta<\omega(1+\varepsilon)\\}$ if $\theta>{\aleph_{0}}$
and is $\\{\alpha_{w,\zeta}:\zeta<1+\varepsilon\\}$,

  4. (d)

$u_{w,v,\varepsilon}$ is included in
$\cup\\{\alpha_{w,\zeta}:\zeta<\omega(1+\varepsilon)\\}$

  5. (e)

$\cup\\{u_{w,v,\varepsilon}:\sigma<\theta\\}=\
v\cap\cup\\{\alpha_{w,\varepsilon}:\varepsilon<\theta\\}$

Lastly we define $\mathbf{U}$ as the set
$\\{\bar{u}_{w,v}:w\in{\mathscr{S}},v\in{\mathscr{A}}_{w}\\}$; so it suffice
to prove that $\mathbf{U}$ witnesses ${\rm MGC}_{D}(\lambda,\theta)=\lambda$;
this is as in previous cases.

2) As in [She94, Ch.III], and anyhow not used .

3) By [She96] there is ${\mathscr{S}}$ such that:

  1. $(*)_{1}$

    1. (a)

${\mathscr{S}}\subseteq[\lambda]^{\theta}$ has cardinality $\lambda$

    2. (b)

${\mathscr{S}}$ is stationary, i.e. for every model $M_{*}$ with universe
$\lambda$ and vocabulary $\leq\theta$ there is $w\in{\mathscr{S}}$ such that
$M_{*}{\restriction}w\prec M_{*}$.

Now as we can increase ${\mathscr{S}}$, without loss of generality :

  1. $(*)_{2}$

${\mathscr{S}}\cap[\alpha]^{\theta}$ is a stationary subset of
$[\alpha]^{\theta}$ for every $\alpha\leq\lambda$.

We continue as in the proof of part (1), maybe details will be given in [S+]
and anyhow this will not be used here.

4) Let $\mathbf{U}\subseteq\mathbf{U}_{\lambda,\theta}$ medium guess clubs,

Now clause (a) follows by 2.1(3)(b).

For clause (b), for $\overline{u}\in\mathbf{U}_{1},$ let
$\langle\alpha_{u,\varepsilon}:\varepsilon<\theta\rangle$ list
$\bigcup_{\varepsilon}u_{\varepsilon}$ and for each $\overline{u}$ and
increasing $g\in{}^{\theta}\theta$ we define
$w_{\overline{u},\varepsilon}^{g}$ by induction on $\varepsilon<\theta$ as
follows. For $\varepsilon=0$ let $w_{\overline{u},\varepsilon}^{g}={\rm
c{\ell}}_{\mathfrak{B}}(\emptyset),$ for $\epsilon$ limit let
$w_{\overline{u},\varepsilon}$ =
$\bigcup\\{w_{\overline{u},\zeta}:\zeta<\epsilon\\},$ so let
$\epsilon=\zeta+1.$ Let $\iota<\theta$ be minimal $\geq g(\zeta),\epsilon,$
such that
$\sup(w_{\overline{u},\zeta}^{g})<\sup\\{\alpha_{\overline{u},\iota(1)}:\iota(1)<i\\}$
and let $w_{\overline{u},\varepsilon}^{g}={\rm
c{\ell}}_{\mathfrak{B}}(\\{\alpha_{\overline{u},\iota(1)}:i(1)<\iota\\}).$
Lastly let $\mathscr{F}\subseteq\\{g\in{{}^{\theta}\theta}:$g$\ \text{is
increasing}\\}$ be $<_{J_{\theta}^{\mathrm{bd}}}-$unbounded of cardinality
$\mathfrak{b}_{\theta}$ and let
$\mathbf{U}=\\{\langle\overline{w}_{\overline{u},\varepsilon}^{g}:\epsilon<\theta\rangle:g\in\mathscr{F}\
\text{and}\ \overline{u}\in\mathbf{U}_{1}\\}.$ Now check.

5), 6) Combine things above. ∎

######  Discussion 2.6.

Assume $\lambda>\theta\geq\sigma={\rm cf}(\sigma),(2^{\sigma}>\lambda$ in the
interesting case). Let
$\mathbf{U}_{\lambda,\theta,\sigma}=\\{\bar{u}:\bar{u}=\langle
u_{\varepsilon}:{\varepsilon}<\sigma\rangle$ is $\subseteq$-increasing and
$u_{\varepsilon}\in[\lambda]^{\theta}\\}$ and repeat the definition. Of
doubtful help, otherwise $(\theta^{++},\theta^{+},\theta)$ would have helped.

######  Theorem 2.7.

1) Assume $\lambda={\rm cf}(\lambda)>\theta={\rm cf}(\theta),\
D=[\theta]^{\theta},\ $ ${\rm AGC}_{D}(\lambda,\theta)=\lambda$ and
${\mathfrak{b}}_{\theta}>\lambda$. Then $\lambda\notin{\rm Univ}(T_{{\rm
ceq}})$; moreover, ${\rm univ}_{T_{{\rm
ceq}}}(\lambda)\geq{\mathfrak{b}}_{\theta}$.

1A) In part (1) we can replace ${\rm AGC}_{D}(\lambda,\theta)$ by ${\rm
MGC}(\lambda,\theta)$.

2) If $\lambda={\rm cf}(\lambda)>\theta={\rm cf}(\theta)$ and888 Recall that
this means that $D=\\{\theta\\}$ ${\rm FGC}(\lambda,\theta)=\lambda$ and
$\chi={\mathfrak{d}\/}_{\theta}>\lambda$ or just ${\rm
cf}({}^{\theta}\theta,\leq_{D}))\geq\chi>\lambda$, then ${\rm univ}_{T_{{\rm
ceq}}}(\lambda)\geq\chi$.

3) If $D$ is a uniform filter on $\theta,({}^{\theta}\theta,<_{D})$ is
$(<\chi)$-directed and $\chi>\lambda,{\rm FGC}_{D}(\lambda,\theta)=\lambda$
then ${\rm univ}_{T_{{\rm ceq}}}(\lambda)\geq\chi$.

######  Remark 2.8.

1) The Claim 2.11 below shows that we cannot weaken the assumption on $T$ too
much.

2) Note that the above works also for $\theta={\aleph_{0}}$.

3) See more in [S+].

###### Proof..

1) So let ($T=T_{{\rm ceq}}$ and):

  1. $(*)_{0}$

$\mathfrak{B}$ is as in Definition 2.1, such that:

    1. (a)

$\mathfrak{B}$ has universe $\lambda,$

    2. (b)

$\tau_{\mathfrak{B}},$ the vocabulary of $\mathfrak{B},$ is countable,

    3. (c)

$\mathfrak{B}$ has a pairing function
$\mathrm{pr}:\lambda\times\lambda\to\lambda$ i.e., a one-to-one 2-$place$
function from $\lambda$ into $\lambda$ and $\mathrm{pr}_{1}$ and
$\mathrm{pr}_{2}$ its inverses.

  2. $(*)_{1}$

assume $\alpha_{*}<{\mathfrak{b}}_{\theta}$ and $M^{*}_{\alpha}\in{\rm
EC}_{T}(\lambda!)$ for $\alpha<\alpha_{*}$; it suffices to find $N\in{\rm
EC}_{T_{{\rm ceq}}}(\lambda!)$ not embeddable into $M^{*}_{\alpha}$ for every
$\alpha<\alpha_{*}$

  3. $(*)_{2}$

let ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ witness ${\rm
AGC}_{D}(\lambda,\theta,\mathfrak{B})=\lambda$ or just ${\rm
MGC}_{D}(\lambda,\theta,\mathfrak{B})=\lambda$.

[Why? Is there such $\mathbf{U}?$ by the assumption of the theorem and apply
2.2(5).]

  4. $(*)_{3}$

for $\bar{u}\in{\mathbf{U}},\alpha<\alpha_{*}$ and $d\in P^{M^{*}_{\alpha}}$
we define the set $E_{\bar{u},d,\alpha}$; clearly is a club of $\theta$, as
follows:

    1. $\bullet$

$E_{\bar{u},d,\alpha}=\\{{\varepsilon}<\theta:{\varepsilon}$ is a limit
ordinal such that $u_{\varepsilon}$ is closed inside
$\cup\\{u_{\zeta}:\zeta<\theta\\}$ under the functions of $M^{*}_{\alpha}$ and
the function $F^{M^{*}_{\alpha}}(-,d)$, or just if $a\in
u_{\varepsilon},b\in\bigcup\limits_{\zeta<\theta}u_{\zeta}$ and
$M^{*}_{\alpha}\models``F^{M^{*}_{\alpha}}(a,d)=b"$ then
$b\in\sup(u_{\varepsilon})\\}$.

So
${\mathscr{E}}=\\{E_{\bar{u},d,\alpha,}:\bar{u}\in{\mathbf{U}},\alpha<\alpha_{*}$
and $d\in P^{M^{*}_{\alpha}}\\}$ is a set of clubs of $\theta$ of cardinality
$\leq|{\mathbf{U}}|+|\alpha_{*}|+|P^{M_{\bullet}}|<{\mathfrak{b}}_{\theta}$.
Hence,

  1. $(*)_{3.1}$

there is an increasing function $g:\theta\rightarrow\theta$ such that
$(\forall
E\in{\mathscr{E}})(\forall^{\infty}{\varepsilon}<\theta)(g({\varepsilon})>{\rm
suc}_{E}({\varepsilon}))$.

Now we can construct $N=N_{g}\in{\rm EC}_{T}(\lambda!)$ such that:

  1. $(*)_{4}$

    1. (a)

$P^{N}=\\{3\beta:\beta<\lambda\\}$ hence
$Q^{N}=\\{3\beta+1,3\beta+2:\beta<\lambda\\}$

    2. (b)

if $\alpha=3\beta+1<\lambda$ (hence $\alpha\in Q^{N}$) then
$\alpha=\min(\alpha/E^{N})$

    3. (c)

if $\bar{u}\in{\mathbf{U}}$ then for some
$\alpha(\bar{u})=\alpha_{\bar{u},g}\in P^{N}$ we have: if
$\beta\in(u_{{\varepsilon}+1}\backslash u_{\varepsilon})\cap Q^{N}$ and
$(\beta/E^{N})\cap u_{\varepsilon}=\emptyset$ but
$\upsilon<\theta\Rightarrow(\beta/E^{N})\cap\sup(\bigcup\limits_{\zeta<\theta}u_{\zeta})\nsubseteq\sup(u_{\upsilon})$
then
$F^{N}(\beta,\alpha(\bar{u}))\in\sup(\bigcup\limits_{\zeta<\theta}u_{\zeta})\backslash\sup(u_{g({\varepsilon}+1)})$.

Now toward contradiction assume that:

  1. $(*)_{5}$

$f$ embeds $N_{g}$ into $M^{*}_{\alpha}$ and $\alpha<\alpha_{*}$.

Let $N^{+}_{g}=(N_{g},<^{N_{g}^{+}})$ where $<^{N_{g}^{+}}\
=\\{(\alpha,\beta):\alpha<\beta<\lambda\\}$ and let $M_{\bullet}$ be a model
with universe $\lambda$ expanding $M^{*}_{\alpha}$ and (a renaming of)
$N^{+}_{g}$; (that is $\tau(M_{\bullet})$ contains also a disjoint copy
$\tau^{\prime}$ of $\tau(N^{+}_{g})$ such that the restriction of
$M_{\bullet}$ to $\tau^{\prime}$ is the suitable copy of $N^{+}_{g}$).

Also we have $f=G^{M_{*}}$ for some unary function symbol
$G\in\tau(M_{\bullet})$ and $M_{\bullet}$ has Skolem functions and
$\tau(M_{\bullet})$ is countable and expand $\mathfrak{B.}$

  1. $(*)_{6}$

    1. (a)

Let $E=\\{\delta<\lambda:M_{\bullet}{\restriction}\delta\prec M_{\bullet}\\},$
so a club of $\lambda,$

    2. (b)

Let $H:\lambda\to\lambda$ be $H(\alpha)={\rm suc}_{E}(\alpha),$

    3. (c)

Let $M_{*}=(M_{\bullet},H).$

  2. $(*)_{7}$

Let $A=\\{\mathrm{pr}^{\mathfrak{B}}(3\beta+1,f(3\beta+1)):\beta<\lambda\\}.$

As $f$ is a one-to-one function from $\lambda$ to $\lambda$ necessarily
$A\in[\lambda]^{\lambda}.$ By the choice of $\mathbf{U}$ and of $D$ as
$[\theta]^{\theta}$ there is $\overline{u}$ such that:

  1. $(*)_{8}$

    1. (a)

$\bar{u}\in{\mathbf{U}}$,

    2. (b)

if $\varepsilon<\theta$ then $u_{\delta}={\rm
c{\ell}}(u_{\varepsilon},\mathfrak{B}),$

    3. (c)

$c\ell(u_{\varepsilon},M_{*})\subseteq\sup(u_{\varepsilon})$, and (essentially
follows) $M_{\bullet}{\restriction}\sup(u_{\varepsilon})\prec M_{\bullet},$

    4. (d)

the set $v=\\{\varepsilon<\theta:A\cap
u_{\varepsilon+1}\setminus\sup(u_{\varepsilon})\not=\emptyset\\}$ has
cardinality $\theta$,

    5. (e)

$M_{*}{\restriction}\bigcup\limits_{{\varepsilon}<\theta}u_{\varepsilon}\prec
M_{*},$ (not used when we assume only ${\rm MGC}(\lambda,\theta)=\lambda$),

Now let $d=f(\alpha_{\bar{u},g})$, so it is a member of $P^{M^{*}_{\alpha}}$,
and

  1. $(*)_{9}$

if $\varepsilon\in v$ then for some $a=a_{\varepsilon}\in
u_{\varepsilon+1}\setminus\sup(u_{\varepsilon})$ we have $a_{\varepsilon}\in
A$,

  1. $(*)_{10}$

    1. (a)

for $\varepsilon\in v$ let
$a_{{\varepsilon}}=\mathrm{pr}^{\mathfrak{B}}(\beta_{\varepsilon},f(\beta_{\varepsilon})),$
so $\beta_{{\varepsilon}}\in\\{3\gamma+1:\gamma<\lambda\\},$

    2. (b)

Let $v_{1}=\\{{\varepsilon}\in v:g({\varepsilon})>{\rm
suc}_{E(\overline{u},d,\alpha)}({\varepsilon})\\},$

    3. (c)

For ${\varepsilon}<\theta$ let $\zeta_{{\varepsilon}}={\rm
suc}_{E(\overline{u},d,\alpha)}({\varepsilon}),$

  2. $(*)_{11}$

$v_{1}\in[v]^{\theta},$

[Why? Because $v\in[\theta]^{\theta}$ and the choice of $f$ (here the use of
$\mathfrak{b}_{\theta}$ matters.)]

Now,

  1. $\boxplus$

Assume $\epsilon\in v_{1},$

  1. $\bullet_{1}$

$a_{{\varepsilon}}\in u_{{\varepsilon}+1}\setminus u_{{\varepsilon}},$

[Why? by the choice of $a_{{\varepsilon}}.$]

  1. $\bullet_{2}$

if $\zeta<\theta$ then $\sup(u_{\zeta})$ is closed under $x+1$ so a limit
ordinal and $f^{-1};$ that is,
$\forall\beta<\lambda[\beta<\sup(u_{{\varepsilon}})\iff
f(\beta)<\sup(e_{{\varepsilon}})],$

[Why? As ${\rm
c{\ell}}(\sup(u_{{\varepsilon}}),M_{*})\subseteq\sup(u_{{\varepsilon}})$ by
$(*)_{8}(c)$.]

  1. $\bullet_{3}$

$\beta_{{\varepsilon}}\in u_{{\varepsilon}+1}\setminus\sup(u_{{\varepsilon}})$
and $f(\beta_{{\varepsilon}})\in
u_{{\varepsilon}+1}\setminus\sup(u_{{\varepsilon}}),$

[Why? As
$\beta_{{\varepsilon}}=\mathrm{pr}_{1}^{\mathfrak{B}}(a_{{\varepsilon}}),f(\beta_{{\varepsilon}})=\mathrm{pr}_{2}^{\mathfrak{B}}(a_{{\varepsilon}})$
and $a_{{\varepsilon}}\in u_{{\varepsilon}+1}$ by $\bullet_{1}$ (and
$u_{{\varepsilon}+1}={\rm c{\ell}}_{\mathfrak{B}}(u_{{\varepsilon}+1})$) by
$(*)_{8}(b)$ we have $\beta_{{\varepsilon}}\in u_{{\varepsilon}+1},\
f(\beta_{{\varepsilon}})\in u_{{\varepsilon}+1}.$ Now,
$\beta_{{\varepsilon}}<\sup(u_{{\varepsilon}})\iff
f(\beta_{{\varepsilon}})<\sup(u_{{\varepsilon}})$ by $\bullet_{2}$ and
$(\beta_{{\varepsilon}},f(\beta_{{\varepsilon}})<\sup(u_{{\varepsilon}}))\Rightarrow
a_{{\varepsilon}}=\mathrm{pr}^{\mathfrak{B}}(\beta_{{\varepsilon}},f(\beta_{{\varepsilon}}))\
\subseteq\sup(u_{{\varepsilon}+1}).$ But,
$a_{{\varepsilon}}\notin\sup(u_{{\varepsilon}}),$ together
$\beta_{{\varepsilon}},f(\beta_{{\varepsilon}})\notin\sup(u_{{\varepsilon}})$
and we are done proving $\bullet_{3}$.]

  1. $\bullet_{4}$

$F^{M_{\alpha}^{*}}(f(\beta_{{\varepsilon}}),d)\notin[\zeta_{{\varepsilon}},\sup\left(\bigcup_{{\varepsilon}}u_{{\varepsilon}}\right)),$

[Why? by the definition of $E_{\overline{u},d,\alpha}$ in $(*)_{3}$.]

  1. $\bullet_{5}$

$F^{M_{\alpha}^{*}}(f(\beta_{{\varepsilon}}),d)=f(F^{N_{g}}(\beta_{{\varepsilon}},\alpha_{\overline{u},g})),$

[Why? As $f$ embed $N_{g}$ into $M_{\alpha}^{*}$ and the choice of $d$ as
$f(\alpha_{\overline{u},g}.)$]

  1. $\bullet_{6}$

$F^{N_{g}}(\beta_{{\varepsilon}},\alpha_{\overline{u},g})\in\left[\sup(u_{g({\varepsilon})}),\sup(\bigcup_{\zeta}u_{\zeta})\right),$

[Why? By the choice of $N_{g},\alpha_{\overline{u},g}$ and $\bullet_{3}.$]

  1. $\bullet_{7}$

$f(F^{N_{g}}(\beta_{{\varepsilon}},\alpha_{\overline{u},g}))\in\left[\sup(u_{g({\varepsilon})}),\sup(\bigcup_{\zeta}u_{\zeta})\right),$

[Why? By $\bullet_{2}$ and $\bullet_{6}.$]

  1. $\bullet_{8}$

$f(F^{N_{g}}(\beta_{{\varepsilon}},\alpha_{\overline{u},g}))\in\left[\sup(u_{\zeta({\varepsilon})}),\sup(\bigcup_{\zeta}u_{\zeta})\right)$
recalling $\zeta({\varepsilon})=J_{{\varepsilon}}={\rm
suc}_{E(\overline{u},d,\alpha)}({\varepsilon}),$

[Why? By $\bullet_{7},$ the definition of $E_{\overline{u},d,\alpha}$ and the
assumption on ${\varepsilon}\in V_{1}.$]

  1. $\bullet_{9}$

$F^{M_{\alpha}^{*}}(f(\beta_{*}),d)\in\left[\zeta_{{\varepsilon}},(\sup(\bigcup_{\zeta}u_{\zeta})\right),$

[Why? By $\bullet_{8}$ and $\bullet_{5}.$]

But $\bullet_{4}$ and $\bullet_{9}$ are contradictory, so we are done proving
part $(1).$

1A), 2), 3) Similarly to the proof of part (1) with some changes will be done
in [S+].

∎

######  Conjecture 2.9.

1) Assume $T$ (is countable complete first order) with the ${\rm TP}_{2}$. If
$\lambda>\theta>\aleph_{0}$ are regular, ${\mathfrak{d}\/}_{\theta}>\lambda$
and ${\mathfrak{d}\/}_{\kappa}>{\rm FGC}(\lambda,\theta)$ (maybe $\theta$
inaccessible), then ${\rm univ}_{T}(\lambda)\geq{\mathfrak{d}\/}_{\kappa}$.

2) Assume $T$ (is countable complete first order) non-simple. If
$\lambda>\theta>\aleph_{0}$ are regular, ${\mathfrak{d}\/}_{\theta}>\lambda$
and ${\mathfrak{d}\/}_{\kappa}>{\rm CFGC}(\lambda,\theta)$, then ${\rm
univ}_{T}(\lambda)\geq{\mathfrak{d}\/}_{\kappa}$.

######  Remark 2.10.

See hopefully [Shec], [S+].

######  Claim 2.11.

Assume $\mu=\mu^{<\mu}\leq\theta={\rm cf}(\theta)<\lambda={\rm
cf}(\lambda)<\chi=\chi^{\lambda}$, $\lambda$ is strongly inaccessible Mahlo
and for transparency ${\rm GCH}$ holds in the interval $[\mu,\chi)$. For some
${\mathbb{P}}$:

  1. (a)

${\mathbb{P}}$ is a $(<\mu)$-complete forcing of cardinality $\chi$ neither
collapsing any cardinal, nor changing cofinalities

  2. (b)

$(2^{\mu})^{{\mathbf{V}}[{\mathbb{P}}]}=\chi$

  3. (c)

in $\mathbf{V}^{\mathbb{P}}$ we have $\mathfrak{d}_{\partial}=\chi$ for every
inaccessible $\partial\in[\mu,\lambda)$,

  4. (d)

in $\mathbf{V}^{\mathbb{P}}$ there is $\bar{{\mathscr{P}}}$ as in 1.5(2),

  5. (e)

$T_{{\rm ceq}}$ has no universal member in $\lambda$ moreover ${\rm
univ}(\lambda,T_{{\rm ceq}})\geq\chi$

  6. (f)

the results of [She90] holds, i.e. there is a universal random graph in
$\lambda$, and see [She21]

######  Remark 2.12.

Note that here the case
${``}\cup_{\varepsilon<\theta}u_{\varepsilon}\cap\sup(u_{\zeta})$ has
cardinality $\theta$” does not arrise.

###### Proof..

Our proof is based on the proof [She90], but the quoted [Bau76] has to be
changed, see full proof in a work by Mark Poór and the author in preparation.

That is, we choose:

  1. ($*$)

$\mathbb{P}=\mathbb{P}_{3}$, where
$\langle\mathbb{P}_{k},\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\ell}:k\leq 3,{\ell}<3\rangle$ is an iteration and:

    1. (A)

$\mathbb{Q}_{0}$ is adding $\chi$ $\lambda$-Cohen, so it satisfies:

      1. $\bullet_{1}$

$\mathbb{Q}_{0}$ is a $(<\lambda)$-complete forcing notion of cardinality
$\chi$,

      2. $\bullet_{2}$

$\mathbb{Q}_{0}$ neither collapse some cardinal nor change any cofinality (in
fact is $\lambda^{+}$-cc),

      3. $\bullet_{3}$

in $\mathbf{V}^{\mathbb{Q}_{0}}$ there is a family ${\mathscr{A}}_{0}$ of
$\chi$-many subsets of $\lambda$ each of cardinality $\lambda$, the
intersection of any two having cardinality $<\lambda$,

    2. (B)

in $\mathbf{V}^{\mathbb{Q}_{0}}=\mathbf{V}^{\mathbb{P}_{1}}$ the forcing
notion $\mathbb{Q}_{1}$ satisfies:

       * $\bullet_{1}$

$\mathbb{Q}_{1}$ is a $(<\mu)$-complete $\lambda$-cc forcing notion of
cardinality $\chi$, (yes, $\lambda$-cc not $\lambda^{+}$-cc),

       * $\bullet_{2}$

$\mathbb{Q}_{1}$ does neither collapses some cardinal nor changes any
cofinality,

       * $\bullet_{3}$

in $\mathbf{V}^{\mathbb{P}_{2}}$ there is a family ${\mathscr{A}}_{1}$ of
$\chi$-many subsets of $\lambda$, each of cardinality $\lambda$, the
intersection of any two having cardinality $<\mu$,

       * $\bullet_{4}$

in $\mathbf{V}^{\mathbb{P}_{2}}$ we have $\mathfrak{d}_{\partial}=\chi$ for
every weakly inaccessible $\partial\in(\mu,\lambda)$,

    3. (C)

in $\mathbf{V}^{\mathbb{P}_{2}}$ we have $\mathbb{Q}_{2}$ which is
$(<\mu)$-complete $\mu^{+}$-cc forcing notion, forcing that there is a
universal graph of cardinality $\lambda$.

Now, why are there such $\mathbb{Q}_{\ell}$-s? For clause (A) use the forcing
of adding $\chi$ $\lambda$-Cohens.

For clause (B) we use $\mathbb{Q}_{1}$ such that

  1. $(*)$

$\mathbb{Q}=\mathbb{Q}^{\prime}_{1}\times\mathbb{Q}^{\prime}_{1}$: starting
with $(\mu,\lambda,\chi,{\mathscr{A}}_{0})$ as above:

    1. (a)

$\mathbb{Q}^{\prime}_{1}$ is the product with Easton support of
$\langle\mathbb{Q}_{1,\theta}:\theta\in S\rangle$ where
$S=\\{\theta:\theta\in(\mu,\lambda)$ is an inaccessible non-Mahlo$\\}$ and for
$\theta\in S,\mathbb{Q}_{1.\theta}$ is the forcing adding $\chi$ many
$\theta$-Cohens,

    2. (b)

$\mathbb{Q}^{\prime\prime}_{2}$ forces a refinement ${\mathscr{A}}_{1}$ of
${\mathscr{A}}_{0}$ to a family as required as in [Bau76] but with Easton
support; so each condition has cardinality $<\lambda$ and has Easton support.

Now why clause (B)$\bullet_{1},\bullet_{2},\bullet_{3}$ holds? As said above
noting that [Bau76, 6.1] use full support while use Easton support

Lastly for clause (C) we apply [She90].

Having constructed $\mathbb{P}=\mathbb{P}_{3}$ we have to check that is is as
required.

Now being $(<\mu)$-complete, of cardinality $\chi$, pedantically of density
$\chi$, is obvious by the properties of the
$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{\ell}$-s. Similarly concerning “no cardinal is collapsed
and no cofinality changed”, so clause (a) of Claim 2.11 holds. Also forcing
the existence of a universal graph of cardinality $\lambda$ holds by the
choice of $\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}_{2}$, so clause (f) of 2.11 holds.

Next, clause (c) there saying $\mathfrak{d}_{\partial}=\chi$ holds because it
obviously holds in $\mathbf{V}^{\mathbb{P}_{2}}$ by the choice of
$\mathchoice{\oalign{$\displaystyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle\mathbb{Q}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}^{\prime}_{1}$ and the later forcing preserve it because it
satisfies the $\mu^{+}$-cc. Now, lastly, why clause (d) of 2.11 holds? First,
in $\mathbf{V}^{\mathbb{P}_{1}}$ we have ${\rm GCH}$ in the interval
$[\mu,\lambda)$ so there is such $\bar{{\mathscr{P}}}$, and
$\mathbb{P}_{3}/\mathbb{P}_{1}$ satisfies the $\lambda$-cc so the old clubs of
$\lambda$ are dense. and this continue to holds in $\mathbf{V}^{\mathbb{P}}$.
Hence the non-existence of a universal model of $T_{{\rm ceq}}$ in $\lambda$,
(holds by 1.5(2)). ∎

######  Question 2.13.

1) Can we for theories $T$ satisfying ${\rm NSOP}_{1}+{\rm TP}_{2}$ get
similar results?

2) Is $T_{{\rm ceq}}$ in some sense minimal non-simple in a suitable family of
theories?

######  Claim 2.14.

Assume $\lambda>\theta={\rm cf}(\theta)>\aleph_{0}$ and
$\lambda=\lambda^{\theta}$ and ${\mathbb{P}}$ is a $\theta$-cc forcing notion.

1) In $\mathbf{V}^{\mathbb{P}}$ there is a reasonable strongly bounding
$\mathbf{U}\subseteq\mathbf{U}_{\lambda,\theta}$ of cardinality $\lambda$
witnessing $\lambda={\rm FGC}(\lambda,\theta,\mathfrak{B})$

2) Assume ${\mathbf{U}}\subseteq{\mathbf{U}}_{\lambda,\theta}$ fully guess
clubs then in ${\mathbf{V}}^{{\mathbb{P}}},{\mathbf{U}}$ still fully guess
clubs.

3) In part (2), if ${\mathbf{U}}$ is reasonable/bounding/strongly
bounding/weakly bounding in ${\mathbf{V}}$, then so it is in
${\mathbf{V}}^{{\mathbb{P}}}$.

######  Remark 2.15.

1) This showed help in consistency results.

2) Similarly for the other versions of guessing clubs from 2.1, but take care
of what is $D$.

3) In 2.14(2) we can replace “fully guess clubs” by a slightly stronger
version of “almost guess clubs” specifically, instead of one set $A$ we have
$\sigma<\theta$ sets $A$.

4) Also In 2.14(2) we can use versions with $D$-s.

###### Proof..

Part (1) follows by parts (2),(3) because in $\mathbf{V}$ there is such
$\mathbf{U}$ by 2.4(1)(C). The point is:

  1. $(*)$

$(A)\Rightarrow(B)$ where:

    1. (A)

if $\chi>\lambda$ and
$\\{\mathchoice{\oalign{$\displaystyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}},{\mathbb{P}},\lambda\\}\cup\\{{\varepsilon}:{\varepsilon}\leq\theta\\}\subseteq
N\prec({\mathscr{H}}(\chi),\in)$ and $\|N\|<\lambda$ and $\mathbb{P}$
satisfies the $\theta^{+}$-cc where
$\Vdash_{{\mathbb{P}}}``\mathchoice{\oalign{$\displaystyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}$ a model with universe $\lambda$ and vocabulary of
cardinality $<\theta$”

    2. (B)

$\Vdash_{{\mathbb{P}}}``\mathchoice{\oalign{$\displaystyle N$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle N$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle N$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle N$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}\cap\lambda={\rm
c{\ell}}(|N|,\mathchoice{\oalign{$\displaystyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}})$ and
$\mathchoice{\oalign{$\displaystyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\restriction}|N|$ is an elementary submodel of
$\mathchoice{\oalign{$\displaystyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\displaystyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\textstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\textstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}{\oalign{$\scriptscriptstyle{\mathfrak{B}}$\crcr\vbox
to0.86108pt{\hbox{$\scriptscriptstyle{\tilde{\mkern-3.0mu}\mkern
3.0mu}{}$}\vss}}}"$.

∎

######  Definition 2.16.

Assuming $\theta={\rm cf}(\theta)\leq\lambda$ we let
$\mathfrak{d}^{\dagger}_{\theta\lambda}$ be the cofinality of the partial
order
$({\mathscr{F}}^{\dagger}_{\theta,\lambda},\leq^{\dagger}_{\theta,\lambda})$
where:

  1. $(*)_{1}$

${\mathscr{F}}^{\dagger}_{\theta,\lambda}$ is the family of subsets of
${}^{\theta}\theta$ of cardinality $\leq\lambda$

  2. $(*)_{2}$

let $\leq=\leq^{\dagger}_{\theta,\lambda}$ is the following partial order on
${\mathscr{F}}\dagger_{\theta,\lambda}$:

$F_{1}\leq F_{2}$ if $(\forall f_{1}\in F_{1})(\exists f_{2}\in
F_{2})[f_{1}\leq f_{2}]$

The following was part of 2.7, maybe we shall return to it.

######  Claim 2.17.

1) If $\lambda>\theta>{\aleph_{0}}$ are regular, ${\rm
FGC}(\lambda,\theta)=\lambda$ and $\mathfrak{d}_{\theta}>\lambda$ then ${\rm
univ}(\lambda,T_{{\rm ceq}})\geq\mathfrak{d}_{\theta}$.

2) Above we can replace $\mathfrak{d}_{\theta}$ by
$\mathfrak{d}^{\dagger}_{\theta,\lambda}$

###### Proof..

1) Like the proof of part (1) of 2.7 so we mainly note the changes.

  1. $(*)_{1}$

as above.

  1. $(*)_{2}$

$\mathbf{U}\subseteq\mathbf{U}_{\lambda,\theta}$ witness ${\rm
FGC}(\lambda,\theta)=\lambda$.

  1. $(*)_{3}$

we let:

    1. (a)

$M^{+}_{\alpha}$ is an expansion of $M^{*}_{\alpha}$ by a pairing function and
Skolem functions,

    2. (b)

${\mathscr{Z}}=\\{(\alpha,\bar{u},d):\alpha<\alpha_{*},\cup_{\varepsilon<\theta}u_{\varepsilon}$
is closed under the function $F^{M^{+}_{\alpha}}$ and each $u_{\varepsilon}$
is closed under the functions of $M^{+}_{\alpha}\\}$,

    3. (c)

for $(\alpha,\bar{u},d)\in{\mathscr{Z}}$ let
$E_{\alpha,\bar{u},d}=\\{\varepsilon<\theta:u_{\varepsilon}$ is closed under
$F^{M^{*}_{\alpha}}(-,d)\\}$,

    4. (d)

above let $g_{\alpha,\bar{u},d}\in{}^{\theta}\theta$ be such that
$g_{\alpha,\bar{u},d}(\varepsilon)=\min(E_{\alpha,\bar{u},d}\setminus(\varepsilon+1))$,
$g\in{}^{\theta}\theta$

Next choose $g\in{}^{\theta}\theta$ not bounded by any well defined
$g_{\alpha,\bar{u},d}$ Now we choose $N=N_{g}$ as follows:

  1. $(*)_{4}$

we let

    1. (a),(b)

as above,

    2. (c)

for every $\bar{u}\in\mathbf{U}$ for some
$\alpha(\bar{u})=\alpha_{\bar{u},v}\in P^{N}$ for every $\varepsilon<\theta$
we have:

       * •

$F^{N}(-,d)$ maps $\cup_{\varepsilon<\theta}u_{\varepsilon}$ into itself

       * •

for $\varepsilon<\theta$ we have: $\varepsilon\in v$ iff there is $a\in
u_{\varepsilon+1}\setminus u_{\varepsilon}$ such that $F^{N}(a,d)\notin
u_{\varepsilon+1}$.

The rest is as in the proof of part (1) of 2.7. ∎

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