$(A_{\alpha})$-type $p(\bar{x})$ that is realized if and only if $f$ has a
multiplicative refinement, and, by assumption, all $(A_{\alpha})$-types are
realized in $\widehat{\mathcal{M}}$.

($\Leftarrow$): Let $p(\bar{x})$ be an $(A_{\alpha})$-type of
$\widehat{\mathcal{M}}$ having cardinality $\lambda$ and choose a set of
representatives $(r_{\beta})_{\beta<\eta}$ for the parameters of $p(\bar{x})$
from $\prod_{\alpha<\lambda}A_{\alpha}$ and notice that these representatives
witness $p(\bar{x})$ being graph-like. Let $f$ be a distribution for
$p(\bar{x})$ using the same representatives. By Corollary 6.16 it must be that
$f$ is a refinement of graph-like distribution $\hat{f}$ with the property
that $(\hat{g}_{n})_{n\in\omega}$, the conjugate of $\hat{f}$, is such that
$\hat{g}_{1}=g_{1}$ (where $g_{1}$ comes from the conjugate of $f$) and the
graphs
$\hat{\mathbb{G}}_{\alpha}:=\langle\hat{g}_{1}(\alpha),\hat{g}_{2}(\alpha)\rangle$
are induced subgraphs of $\langle k_{1}(\alpha),k_{2}(\alpha)\rangle$ where
$(k_{n})_{n\in\omega}$ is the Łoś conjugate of $p(\bar{x})$. In particular,
for all $\alpha\in\mathrm{ess}(f)$ we know that $\hat{\mathbb{G}}_{\alpha}$ is
isomorphic to a graph arising in $\mathrm{Sh}(A_{\alpha})$ by Lemma 7.5. Lemma
7.15 guarantees that $\hat{f}$ satisfies (7.2), so $\hat{f}$ has a
multiplicative refinement by assumption. Thus, by Fact 2.9, the type
$p(\bar{x})$ is realized. ∎

######  Theorem 7.17.

Let $\mathcal{D}$ be a regular ultrafilter on the infinite cardinal $\lambda$,
$A$ be a graph-like $\varphi$-set in the structure $\mathcal{M}$, and $\Gamma$
be the set of $\leq$-minimal graph isomorphism types not occurring in
$\mathrm{isoSh}(A)$. Then the following are equivalent:

  1. (1)

Every $(A)$-type of $\widehat{\mathcal{M}}:=\mathcal{M}^{\mathcal{D}}$ with
cardinality $\leq\lambda$ is realized.

  2. (2)

If $(\mathbb{G}_{\alpha})_{\alpha<\lambda}$ is a sequence of (possibly
infinite) graphs with the property that for every $\gamma\in\Gamma$ and
$\alpha<\lambda$ we have that
$\gamma\nleq\mathrm{isoType}(\mathbb{G}_{\alpha})$ (i.e. every finite induced
subgraph of $\mathrm{isoType}(\mathbb{G}_{\alpha})$ is an element of
$\mathrm{isoSh}(A)$), then the ultraproduct
$\mathbb{G}:=\mathbb{G}_{\alpha}^{\mathcal{D}}$ has the property that every
complete $H\subseteq\mathbb{G}$ with $|H|=\lambda$ extends to an internal
complete subgraph of $\mathbb{G}$.

###### Proof.

$(1)\Rightarrow(2)$: We proceed by constructing an $(A)$-type $p(\bar{x})$ in
$\widehat{\mathcal{M}}$ by taking advantage of the fact that the isomorphism
types of the factors of $\mathbb{G}$ belong to $\mathrm{isoSh}(A)$. We then
show that a multiplicative distribution for $p(\bar{x})$ in
$\widehat{\mathcal{M}}$ gives instructions for extending $H$ to an internal
complete subgraph of $\mathbb{G}$.

For each $\alpha<\lambda$ choose a pair
$(\mathbb{G}^{\prime}_{\alpha},\theta_{\alpha})$ in $\mathrm{Sh}(A)$ with a
graph isomorphism
$\psi_{\alpha}\colon\mathbb{G}_{\alpha}\to\mathbb{G}^{\prime}_{\alpha}$.
Enumerate the elements of $H$ as $(h_{\beta})_{\beta<\lambda}$ and pick
representatives $x_{\beta}\in\prod_{\alpha<\lambda}\mathbb{G}_{\alpha}$ for
each $h_{\beta}$. For each $\alpha,\beta<\lambda$, define

$\bar{r}_{\beta}(\alpha)=\theta_{\alpha}(\psi_{\alpha}(x_{\beta}(\alpha)))$

and consider the collection of formulas
$p(\bar{x})=\\{\varphi(\bar{x},\bar{r}_{\beta}/\mathcal{D}):\beta<\lambda\\}$.
We claim that $p(\bar{x})$ is an $(A)$-type of $\widehat{\mathcal{M}}$. It
will be enough to check that $p(\bar{x})$ is a type as the image of
$\theta_{\alpha}$ is always a subset of $A$.

If $\Delta\in\mathcal{P}_{\omega}(\lambda)$, then, because $A$ is graph-like,
we have that

$\displaystyle\llbracket\Delta\rrbracket^{\mathcal{D}}_{p(\bar{x})}$
$\displaystyle=\\{\alpha\in\lambda:\forall\\{\beta,\gamma\\}\in[\Delta]^{2},\,\mathcal{M}_{\alpha}\vDash\exists\bar{x},\,\varphi(\bar{x},\bar{r}_{\beta}(\alpha))\wedge\varphi(\bar{x},\bar{r}_{\gamma}(\alpha))\\}$
$\displaystyle=\\{\alpha\in\lambda:\forall\\{\beta,\gamma\\}\in[\Delta]^{2},\,\mathbb{G}^{\prime}_{\alpha}\vDash\psi_{\alpha}(x_{\beta}(\alpha))\mathrel{E}\psi_{\alpha}(x_{\gamma}(\alpha))\\}$
$\displaystyle=\\{\alpha\in\lambda:\forall\\{\beta,\gamma\\}\in[\Delta]^{2},\,\mathbb{G}_{\alpha}\vDash
x_{\beta}(\alpha)\mathrel{E}x_{\gamma}(\alpha)\\}$
$\displaystyle=\bigcap_{\\{\beta,\gamma\\}\in[\Delta]^{2}}\\{\alpha\in\lambda:\mathbb{G}_{\alpha}\vDash
x_{\beta}(\alpha)\mathrel{E}x_{\gamma}(\alpha)\\}$

Then, as $x_{\beta}/\mathcal{D}=h_{\beta}$ and
$x_{\gamma}/\mathcal{D}=h_{\gamma}$ are neighbors in $\mathbb{G}$ and
$[\Delta]^{2}$ is finite, we have that this last set is an element of
$\mathcal{D}$ by Łoś theorem and the fact that ultrafilters are closed under
finite intersection.

By assumption, the type $p(\bar{x})$ is realized in $\widehat{\mathcal{M}}$
so, by Fact 2.9, we may choose a multiplicative distribution $f$ for
$p(\bar{x})$ and we may assume that $\mathrm{ess}(f)=\lambda$ (if not, let
$f^{\prime}(\Delta)=f(\Delta)$ if $\Delta\neq\\{0\\}$ and set
$f^{\prime}(\\{0\\})=f(\\{0\\})\cup(\lambda\setminus\mathrm{ess}(f))$. Then
$f^{\prime}$ is a multiplicative distribution of $p(\bar{x})$ and
$\mathrm{ess}(f^{\prime})=\lambda$). We will use the information contained in
$f$ to extend $H$.

Let $(g_{n})_{n\in\omega}$ be the conjugate of $f$. For each $\alpha<\lambda$,
define $\mathbb{H}_{\alpha}$ to be the subgraph of $\mathbb{G}_{\alpha}$
induced by the set

$\psi_{\alpha}^{-1}[\\{\psi_{\alpha}(x_{\beta}(\alpha)):\beta\in
g_{1}(\alpha)\\}]=\\{x_{\beta}(\alpha):\beta\in g_{1}(\alpha)\\}.$

Because $\psi_{\alpha}$ and $\psi^{-1}_{\alpha}$ are graph isomorphisms and
because the graph $\langle g_{1}(\alpha),g_{2}(\alpha)\rangle$ is complete by
Lemma 4.5, we must have that $\mathbb{H}_{\alpha}$ is complete for all
$\alpha<\lambda$. Thus
$\widehat{\mathbb{H}}:=\prod_{\alpha<\lambda}\mathbb{H}_{\alpha}/\mathcal{D}$
is complete by Łoś theorem and is internal by construction. Furthermore,
$\widehat{\mathbb{H}}$ extends $H$ as
$x_{\beta}(\alpha)\in\mathbb{H}_{\alpha}$ whenever $\alpha\in f(\\{\beta\\})$
and $f(\\{\beta\\})\in\mathcal{D}$.

$(2)\Rightarrow(1)$: Let $p(\bar{x})$ be an $(A)$-type of
$\widehat{\mathcal{M}}$ of cardinality $\lambda$. We then have that
$p(\bar{x})$ is a graph-like type, witnessed by representations of the
parameters coming from $A^{\lambda}$. The type $p(\bar{x})$ is realized in
$\widehat{\mathcal{M}}$ if and only if the Łoś ultragraph $\mathbb{G}_{L}$ has
the property that the complete subgraph induced by $\eta_{L}[\lambda]$ extends
to an internal complete subgraph of $\mathbb{G}_{L}$ (see Theorem 6.14). ∎

##  8\. Applications to the $\mathrm{SOP}$ Hierarchy

###  8.1. $\mathrm{SOP}_{2}$ and Comparability in Trees

We know from the definition of $\mathrm{SOP}_{2}$ (Definition 2.13) that
whenever $A$ is a $\varphi$-set whose elements form an $\mathrm{SOP}_{2}$-tree
within the structure $\mathcal{M}$ then any $B\subseteq A$ creates a
consistent collection of formula

$p_{B}(\bar{x})=\\{\varphi(\bar{x},\bar{b}):\bar{b}\in B\\}$

if and only if $B$ is a subset of a branch of the $\mathrm{SOP}_{2}$-tree
whose elements come from $A$. In other words, $p_{B}(\bar{x})$ is consistent
if and only if every pair $\bar{b},\bar{b}^{\prime}\in B$ is comparable in the
$\mathrm{SOP}_{2}$-tree (that is, $B$ is a linear sub-order of the tree). We
thus have that the $\varphi$-set $A$ is graph-like and the shape of $A$ is the
collection of all finite graphs $\mathbb{G}$ that can be obtained from taking
a finite (multi-)subset of the full binary tree. There are edges between the
two distinct vertices $b$ $b^{\prime}$ in $\mathbb{G}$ if and only if $b$ and
$b^{\prime}$ are comparable in the full binary tree. Fortunately, most of the
combinatorial work of determining the isomorphism types of such graphs has
already been done by [7].

######  Definition 8.1.

We say that a tuple $(\varphi,A)$ _witnesses $\mathrm{SOP}_{2}$ in a structure
$\mathcal{M}$_ if there is an $\mathrm{SOP}_{2}$-tree

$(\varphi(\bar{x};\bar{y}),(\bar{a}_{\eta})_{\eta\in 2^{<\omega}})$

in $\mathcal{M}$ such that $A=\\{a_{\eta}:\eta\in 2^{<\omega}\\}$.

######  Lemma 8.2.

Suppose that $\varphi$ is a formula and $A$ is a $\varphi$-set such that the
pair $(\varphi,A)$ witnesses $\mathrm{SOP}_{2}$ in the structure
$\mathcal{M}$. Then $A$ is a graph-like $\varphi$-set and the set $\Gamma$ of
$\leq$-minimal isomorphism types not in $\mathrm{isoSh}(A)$ are the two
isomorphism types:

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###### Proof.

Because $(\varphi,A)$ witnesses $\mathrm{SOP}_{2}$ in $\mathcal{M}$, there is
a surjection $h:2^{<\omega}\to A$ with the property that a subset $B$ of
$2^{<\omega}$ is linearly ordered if and only if the collection of formulas

$p_{B}(\bar{x})=\\{\varphi(\bar{x},h(s)):s\in B\\}$

is consistent in $\mathcal{M}$. If $s\in 2^{<\omega}$, we will denote $h(s)$
by $\bar{a}_{s}$. Let $\Delta\in\mathcal{P}_{\omega}(2^{<\omega})$. Then the
set of formula $\\{\varphi(\bar{x},\bar{a}_{s}):s\in\Delta\\}$ is consistent
(equivalently, realized) in $\mathcal{M}$ if and only if the set $\Delta$ is
linearly ordered as a suborder of $2^{<\omega}$ if and only if the elements of
$\Delta$ are pair-wise comparable in the ordering on $2^{<\omega}$. Thus the
elements $(\mathbb{G},\theta)$ of $\mathrm{Sh}(A)$ where $\theta\colon G\to A$
is injective are such that $\mathbb{G}$ is the graph of the comparability
relation for a finite suborder of $2^{<\omega}$ (equivalently, $\mathbb{G}$ is
the graph of the comparability relation for a finite forest of subtrees of
$2^{<\omega}$; i.e. the connected components of $\mathbb{G}$ correspond to
finite subtrees of $2^{<\omega}$). Every finite collection of finite trees can
be embedded as a suborder of $2^{<\omega}$, so the shape of $A$ contains every
possible disjoint union of finitely many comparability graphs arising from
finite trees. We claim that no other isomorphism types appear in
$\mathrm{isoSh}(A)$. We postpone the proof of the claim to study the structure
of these isomorphism types in more detail.

By [7] a finite connected graph $\mathbb{G}=(G,E)$ is the comparability graph
of a tree if and only if for all distinct $x_{0},x_{1},x_{2},x_{3}\in G$ the
formula

$x_{0}\mathrel{E}x_{1}\mathrel{E}x_{2}\mathrel{E}x_{3}\rightarrow(x_{0}\mathrel{E}x_{2})\vee(x_{1}\mathrel{E}x_{3})$

holds (called _the diagonal property_). Clearly, every graph $\mathbb{G}$ with
$|G|\leq 3$ satisfies the diagonal property. Suppose that a graph with
vertices $\\{0,1,2,3\\}$ does not satisfy the diagonal property. We may
suppose that $0\mathrel{E}1\mathrel{E}2\mathrel{E}3$. If these are the only
edges, the graph is $\ell_{4}$ and clearly not diagonal. Now there are only 3
possible edges that can be added, $\\{0,3\\},\\{1,3\\},$ or $\\{0,2\\}$. If
just $\\{0,3\\}$ is added, we have a graph isomorphic to $C_{4}$, which is not
diagonal. We wish to show that the remaining possibilities are all diagonal.
The complete graph on 4 elements is the comparability graph of a tree that is
a 4 element linear order. Up to isomorphism, the remaining possibilities are
the comparability graphs of the two trees

and so are diagonal.

If $\mathbb{G}$ is a non-diagonal graph with greater than $4$ vertices, there
must be a 4-vertex induced subgraph of $\mathbb{G}$ that is not diagonal (pick
the four vertices to be a set of vertices witnessing the fact that the graph
is not diagonal). Thus a graph $\mathbb{G}$ is diagonal if and only if
$\mathbb{G}$ is the disjoint union of a collection of comparability graphs of
trees if and only if $C_{4}\nleq\mathbb{G}$ and $\ell_{4}\nleq\mathbb{G}$.

It remains to check that $\mathrm{isoSh}(A)$ consists of precisely the finite
graph isomorphism types that are diagonal. We already know that the
$(\mathbb{G},\theta)\in\mathrm{Sh}(A)$ with $\theta$ injective have the
property that $\mathbb{G}$ is diagonal (such graphs are the comparability
graphs of a finite subtree of the full binary tree). Suppose that
$(\mathbb{G},\theta)\in\mathrm{Sh}(A)$ and $\theta$ is not injective. Suppose
that $\mathbb{G}$ is not diagonal. As before, we need only look at a set of
vertices $V:=\\{v_{0},v_{1},v_{2},v_{3}\\}$ that witness the failure of the
diagonal property. In particular, $V$ must have the property that the subgraph
of $\mathbb{G}$ induced by $V$ is either $C_{4}$ or $\ell_{4}$ as these are
the only non-diagonal 4-vertex graphs. Because
$(V,\theta\text{\,}\mathord{\upharpoonright}\text{\,}V)$ is an element of
$\mathrm{Sh}(A)$ and because $V$ is diagonal,
$\theta\text{\,}\mathord{\upharpoonright}\text{\,}V$ is not injective. Suppose
that $\theta(v_{i})=\theta(v_{j})$ for a distinct pair of indices
$i,j\in\\{0,1,2,3\\}$. Then the two vertices $v_{i}$ and $v_{j}$ share the
same neighbors (other than each other) and are neighbors of each other by the
definition of $\mathrm{Sh}(A)$. One can check that no distinct pair of
vertices in $C_{4}$ or $\ell_{4}$ satisfy this condition. Thus $\mathbb{G}$ is
diagonal. ∎

In order to use the information in Lemma 7.15 to characterize the graph-like
distributions of $\mathrm{SOP}_{2}$-types, we will need to determine the
necessary sets for the two graph isomorphism types in $\mathrm{isoSh}(A)$.

######  Lemma 8.3.

With $\varphi$, $A$, and $\Gamma$ as in Lemma 8.2, the $\subseteq$-minimal
necessary sets for the elements of $\Gamma$ in $\mathrm{Sh}(A)$ are the edges:

and

where the dotted edges represent the elements of the corresponding necessary
sets.

###### Proof.

By the diagonal characterization of graphs arising from the comparability
relation on finite trees in [7], the analysis in the proof of Lemma 8.2, and
Lemma 7.15, the above sets are necessary for $\ell_{4}$ and $C_{4}$. Since
both $C_{4}$ and $\ell_{4}$ become diagonal when any one of the dotted edges
is added, these necessary sets are $\subseteq$-minimal. ∎

This analysis allows us to give both a characterization of the distributions
of $\mathrm{SOP}_{2}$-types as well as giving a new characterization of good
ultrafilters in terms of which complete subgraphs of ultragraphs with factors
coming from $\mathrm{Sh}(A)$ can be extended to _internal_ complete subgraphs.

######  Theorem 8.4.

Let $\mathcal{D}$ be a regular ultrafilter on the infinite cardinal $\lambda$.
Then the following are equivalent:

  1. (1)

$\mathcal{D}$ is good.

  2. (2)

If $\mathcal{M}$ is a structure and $(\varphi,A)$ witness $\mathrm{SOP}_{2}$
in $\mathcal{M}$ then every $(A)$-type of cardinality $\leq\lambda$ in
$\widehat{\mathcal{M}}:=\mathcal{M}^{\mathcal{D}}$ is realized in
$\widehat{\mathcal{M}}$.

  3. (3)

For all sequences of graphs $(\mathbb{G}_{\alpha})_{\alpha<\lambda}$ such that
$C_{4}\nleq\mathbb{G}_{\alpha}$ and $\ell_{4}\nleq\mathbb{G}_{\alpha}$ for all
$\alpha<\lambda$ the ultraproduct
$\mathbb{G}:=\mathbb{G}_{\alpha}^{\mathcal{D}}$ has the property that every
complete $H\subseteq\mathbb{G}$ with $|H|=\lambda$ is contained within an
internal complete subgraph of $\mathbb{G}$.

  4. (4)

Every graph-like function $f:P_{\omega}(\lambda)\to\mathcal{D}$ that satisfies

$f(\\{x_{0},x_{1}\\})\cap f(\\{x_{1},x_{2}\\})\cap
f(\\{x_{2},x_{3}\\})\subseteq f(\\{x_{0},x_{2}\\})\cup f(\\{x_{1},x_{3}\\})$

for all distinct $x_{0},x_{1},x_{2},x_{3}\in\lambda$ has a multiplicative
refinement.

###### Proof.

$(1)$$\iff$$(2)$: This is a restatement of the fact that a regular ultrafilter
$\mathcal{D}$ is good if and only if $\mathcal{D}$ realizes all
$\mathrm{SOP}_{2}$-types over sets of size $\lambda$, which is proved by
Malliaris and Shelah in [3] (see Conclusion 11.9 and Main Theorem 11.11).

$(2)$$\iff$$(3)$: This is a special case of Theorem 7.17.

$(2)$$\iff$$(4)$: Combining Theorem 7.16 and Lemma 8.3 we have that every
$(A)$-type of $\widehat{\mathcal{M}}$ of cardinality $\lambda$ is realized in
$\widehat{\mathcal{M}}$ if and only if both

$\displaystyle f(\\{x_{0},x_{1}\\})\cap f(\\{x_{1},x_{2}\\})\cap
f(\\{x_{2},x_{3}\\})$ $\displaystyle\subseteq f(\\{x_{0},x_{2}\\})\cup
f(\\{x_{1},x_{3}\\})$ $\displaystyle f(\\{x_{0},x_{3}\\})\cap
f(\\{x_{0},x_{1}\\})\cap f(\\{x_{1},x_{2}\\})\cap f(\\{x_{2},x_{3}\\})$
$\displaystyle\subseteq f(\\{x_{0},x_{2}\\})\cup f(\\{x_{1},x_{3}\\})$

whenever $x_{0},x_{1},x_{2},x_{3}$ are distinct elements of $\lambda$. As the
intersection on the left-hand side of the second condition is always a subset
of the left-hand side of the first condition, both conditions are satisfied if
and only if the first condition is satisfied. ∎

######  Remark 8.5.

The condition on $f$ in statement $(4)$ of Theorem 8.4 is a direct translation
of requiring that the distribution graph sequence for $f$ consists of graphs
satisfying the diagonal property. That is, by Lemma 7.9, statement $(4)$
requires that the distribution graph sequence of $f$ consists of induced
subgraphs of the Łoś graph sequence for some $\mathrm{SOP}_{2}$-type in
$\widehat{\mathcal{M}}$.

###  8.2. Cuts in Linear Orders and $\mathrm{SOP}$

We make an analysis of types that witness a cut in an ultrapower of an
infinite linear order being filled. This analysis follows the same basic
structure as the analysis of $\mathrm{SOP}_{2}$-types presented above, but is
slightly complicated by the shape corresponding to such types being more
complicated. We first characterize the graphs $\mathbb{G}$ whose vertices are
open intervals in some linear order $\mathbb{L}$ and which have an edge
between two intervals $I_{1}$ and $I_{2}$ if and only if $I_{1}\cap I_{2}$ is
nonempty in $\mathbb{L}$. Such graphs will be the elements of the shapes of
the types being studied. Conveniently, the isomorphism types of these graphs
will not depend on the linear order under consideration (see Lemma 8.6 below).
Because of this, we will call such graphs interval intersection graphs without
any reference to the linear order from which the intervals come from.

######  Lemma 8.6.

Suppose that $\mathbb{L}=(L,<)$ is an infinite linear order and that
$\varphi(x;a,b)$ is the formula $a<x<b$. Then $L^{2}$ is a graph-like
$\varphi$-set and the set $\Gamma$ of $\leq$-minimal graphs not in
$\mathrm{Sh}(L^{2})$ consists exactly of the graphs/graph families:

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where $\ell\geq 4$, $m\geq 2$, and $n\geq 1$ (in $\mathrm{V}_{1}$ the nodes
labeled $1$ and $n$ are the same node).

###### Proof.

That $L^{2}$ is a graph-like $\varphi$-set follows from the fact that
$\\{\varphi(x;a_{i},b_{i}):i\in\\{0,1,\dots,n\\}\\}$ is consistent if and only
if the interval $(\max_{i}a_{i},\min_{j}b_{j})$ is non-empty and that this is
equivalent to the intersection $(a_{i},b_{i})\cap(a_{j},b_{j})$ being nonempty
for all $i,j\in\\{0,1,\dots,n\\}$.

The statement of the lemma is now equivalent to asking that the above graphs
are the $\leq$-minimal graphs that cannot arise in the following way: the
vertices are a finite collection of possibly repeated non-empty open intervals
in $\mathbb{L}$ and there is an edge between two intervals if and only if
their intersection is non-empty.

This result (with _distinct_ intervals) is proved in the particular case that
$\mathbb{L}=\mathbb{R}$ by Lekkekerker and Boland [2]. Our strategy will be to
reduce the general case to the case of distinct intervals in $\mathbb{R}$.
That is, we show that $\mathrm{isoSh}(\mathbb{L}^{2})$ is equal to the set of
isomorphism types of interval intersection graphs in $\mathbb{R}$ arising from
distinct intervals.

First, we note that we may assume that $\mathbb{L}$ contains $(\omega,<)$ as a
suborder (if not, the reverse order on $\mathbb{L}$ works). Suppose that
$\mathbb{G}$ is the intersection graph for the collection of open intervals
$\\{(a_{i},b_{i}):i\in\\{0,1,\dots,n\\}\\}$ where $a_{i},b_{i}\in\mathbb{R}$.
We may then construct a set of intervals in $\omega$ by stretching out all of
the $(a_{i},b_{i})$ and translating them to the positive half of $\mathbb{R}$
so that the images of $(a_{i},b_{i})$ and $(a_{j},b_{j})$ intersect in
$\omega$ if and only if $(a_{i},b_{i})\cap(a_{j},b_{j})\neq\emptyset$ and
hence our new intervals have an intersection graph isomorphic to the
intersection graph of the $(a_{i},b_{i})$. Mapping these intervals to
intervals of $\mathbb{L}$ via the inclusion $\omega\hookrightarrow\mathbb{L}$
does not change the intersection graph so long as we did not allow
intersections to look like $(n,n+1)$ in $\omega$ (this can be achieved by
choosing how much to stretch out the original intervals in $\mathbb{R}$). Thus
the set of intersection graphs in $\mathbb{L}$ contains all of the
intersection graphs in $\mathbb{R}$.

For the other inclusion, suppose that
$\mathbb{G}=(\\{I_{i}:i\in\\{0,1,\dots,n\\}\\},E)$ is the intersection graph
for the (possible repeating) non-empty intervals $I_{i}$ in $\mathbb{L}$. Let
$I_{i}=(a_{i},b_{i})$ and, for each $\\{I_{i},I_{j}\\}\in E$, choose an
element $e_{ij}\in I_{i}\cap I_{j}$. Define an order $\mathbb{M}$ with
underlying set the union of the sets

$\displaystyle\\{a_{i}:i\in\\{0,1,\dots,n\\}\\}$
$\displaystyle\\{b_{i}:i\in\\{0,1,\dots,n\\}\\}$
$\displaystyle\\{e_{ij}:i<j,\\{I_{i},I_{j}\\}\in E\\}$

and with ordering given by $<^{\mathbb{L}}$. $\mathbb{M}$ is finite and so
isomorphic to an initial segment of $\omega$. The map
$\mathbb{M}\hookrightarrow\omega\hookrightarrow\mathbb{R}$ sends the intervals
$I_{i}$ to intervals of $\mathbb{R}$ having the same intersection graph as the
$I_{i}$. If the intervals $I_{i_{0}},\dots,I_{i_{m}}$ are equal (and none of
the other intervals are equal to $I_{i_{0}}$), then we may take the distinct
intervals

$\left\\{\left(a_{i_{0}}+\frac{j}{m+1},b_{i_{0}}-\frac{j}{m+1}\right):j\in\\{0,\dots,m\\}\right\\}$

to be the intervals corresponding to the $I_{i_{j}}$ without changing the
intersection graph. ∎

For the next several propositions we fix the following suppositions:
$\mathcal{M}$ is a structure with an infinite $L\subseteq M$ (not necessarily
definable!) and a definable binary relation $<$ on $\mathcal{M}$ such that $<$
restricts to a linear ordering on $L$ that contains a copy of $\omega$ (as an
ordered set). Let $\varphi(x;y,z)$ be the formula $y<x<z$.

The following fact from [2, Theorem 3] will be useful for determining minimal
necessary sets for the graphs that cannot arise as the intersection graph of a
finite collection of intervals.

######  Definition 8.7.

A graph $\mathbb{G}$ has an _irreducible cycle of length $n$_ if there are $n$
distinct vertices $v_{1},\dots,v_{n}$ of $\mathbb{G}$ such that
$v_{1}v_{2}\ldots v_{n-1}v_{n}v_{1}$ is a path and there is an edge between
$v_{i}$ and $v_{j}$ if and only if $i=j\pm 1\pmod{n}$ .

######  Fact 8.8.

A finite graph $\mathbb{G}$ is not able to be represented as the intersection
graph of a finite collection of intervals if and only if it has at least one
of the two following properties:

  1. (1)

$\mathbb{G}$ has an irreducible cycle of length $\geq 4$.

  2. (2)

$\mathbb{G}$ has a collection of three vertices that are pairwise not
neighbors and, for each of the three vertices $v$, there is a path $P$ between
the other two vertices such that $v$ is not a neighbor of any vertex in $P$.

######  Definition 8.9.

A graph $\mathbb{G}$ is called _asteroidal_ if it satisfies the condition of
Fact 8.8(2). If $\mathbb{G}$ is a graph and the vertices
$A=\\{v_{1},v_{2},v_{3}\\}$ witness $\mathbb{G}$ being asteroidal, we will say
that $A$ is an _asteroidal set in $\mathbb{G}$_.

We will use the notion of asteroidal sets to help us to determine the
necessary sets for the minimal asteroidal graphs given in Lemma 8.6. In
particular, we know that if $A$ is asteroidal in the graph $\mathbb{G}$ and
$\mathbb{G}^{\prime}$ is an interval intersection graph extending $\mathbb{G}$
then there must be some $a\in A$ and $g\in\mathbb{G}$ such that $a$ and $g$
are neighbors in $\mathbb{G}^{\prime}$ but not in $\mathbb{G}$ (see Lemma
8.10). There does not appear to be a nice way other than this to determine
what the necessary sets are, so the proofs determining the minimal necessary
sets will largely proceed by ruling out all of the elements not in the
necessary set one at a time.

######  Lemma 8.10.

Let $\mathbb{G}$ be an asteroidal graph and $A=\\{v_{1},v_{2},v_{3}\\}$ be
asteroidal in $\mathbb{H}$ with paths $P_{v_{i}}$ between the nodes
$A\setminus\\{v_{i}\\}$ for each $1\leq i\leq 3$ witnessing $A$ being
asteroidal in $\mathbb{G}$. Then any interval intersection graph
$\mathbb{G}^{\prime}$ having $\mathbb{G}$ as a subgraph must induce a new edge
in $\mathbb{G}$ that is in the set $\bigcup_{i}(\\{v_{i}\\}\times[P_{v_{i}}])$
where $[P_{v_{i}}]\subseteq G$ is the set of vertices in the path $P_{v_{i}}$.

###### Proof.

If $\mathbb{G}^{\prime}$ does not induce any new edges between elements of $A$
and the paths $P_{v_{i}}$, the $P_{v_{i}}$ will also witness the fact that $A$
is asteroidal in $\mathbb{G}^{\prime}$, a contradiction. ∎

We start by analyzing the class of graphs labeled $\mathrm{III}_{\ell}$
because knowing the necessary sets for this class of graphs will be useful in
finding the minimal necessary sets for all of the other classes.

######  Lemma 8.11.

The edges $A_{\ell}:=\\{\\{1,3\\}\\}\cup\\{\\{2,k\\}:4\leq k\leq\ell\\}$, or
any translation of $A_{\ell}$ by an automorphism of $\mathrm{III}_{\ell}$,
forms a $\subseteq$-minimal necessary set for $\mathrm{III}_{\ell}$ in
$L^{2}$.

###### Proof.

That this set is necessary is proven by Lekkekerker and Boland in [2, Lemma 1]
(note that the word _acyclic_ in the referenced lemma means that the graph has
no irreducible cycles of length _greater than 3_ , so every interval
intersection graph is acyclic in this sense). It will then be enough to find
an interval intersection graph $\mathbb{G}$ for each $e\in A_{\ell}$ such that
$\mathbb{G}$ extends $\mathrm{III}_{\ell}$, $e$ is an edge of $\mathbb{G}$,
and no other $e^{\prime}\in A_{\ell}$ is an edge in $\mathbb{G}$.

Let $e=\\{i,j\\}$ where $i\in\\{1,2\\}$ and consider the graph $\mathbb{G}$
that is $\mathrm{III}_{\ell}$ with all of the edges $\\{j,k\\}$ for $k\neq j$
added. If $e=\\{1,3\\}$, no new edges incident to $2$ are added and if $j\neq
3$ then no elements of $A_{\ell}$ are added except $e$. We claim that
$\mathbb{G}$ is an interval intersection graph.

Since $j$ is a neighbor of every element of $\mathbb{G}$, we may construct a
collection of intervals that have an intersection graph isomorphic to the
subgraph of $\mathbb{G}$ induced by $G\setminus\\{j\\}$ and then take $j$ to
be the union of these intervals. The induced subgraph looks like

$(j+1)$$(j+2)$$\ell$$1$$2$$(j-1)$

which is the intersection graph of a collection of intervals that barely
overlap at the endpoints.

The definition of necessary sets for $\mathbb{G}$ in $S$ implies that any
translation of a necessary set for $\mathbb{G}$ by an automorphism is a
necessary set (in particular, composition with an automorphism of $\mathbb{G}$
is a permutation of the injective graph homomorphisms from $\mathbb{G}$ to any
other graph). ∎

######  Lemma 8.12.

The 6 edges (the two dotted edges below and their images under permutation of
the ‘arms’) shown below form a $\subseteq$-minimal necessary set for
$\mathrm{I}$ in $L^{2}$.

###### Proof.

For $0\leq n\leq 3$, we will call the vertices along the ‘arm’ at the angle
$\pi/2+2n\pi/3$ by $a_{n1}$ and $a_{n2}$ where $a_{n1}$ is the neighbor of the
center vertex and $a_{n2}$ is the outermost vertex on the given arm. We will
denote the center vertex by $c$. We then have that the set $A=\\{a_{n2}:0\leq
n\leq 2\\}$ is asteroidal in $\mathrm{I}$. By Lemma 8.10, any intersection
graph having $\mathrm{I}$ as a subgraph must add an edge incident to one of
the elements of $A$, so we will start our search for necessary edges here.

Let $\mathbb{G}$ be an intersection graph and $i:\mathrm{I}\to\mathbb{G}$ an
injective graph homomorphism (we will denote $i(a_{kj})$ by
$a^{\prime}_{kj}$). We may assume that $a^{\prime}_{02}$ has an edge that does
not occur in $\mathbb{G}$ since any permutation of $A$ can be achieved by a
graph automorphism of $\mathrm{I}$. If
$a^{\prime}_{02}\mathrel{E}^{\mathbb{G}}a^{\prime}_{12}$ then the set of
vertices
$\\{a^{\prime}_{12},a^{\prime}_{02},a^{\prime}_{01},c^{\prime},a^{\prime}_{11}\\}$
can be taken to form a subgraph of $\mathbb{G}$ isomorphic to
$\mathrm{III}_{5}$ with the nodes labeled in the order given. From the
necessary sets found for $\mathrm{III}_{n}$, one of the edges
$e_{1}:=\\{a^{\prime}_{02},a^{\prime}_{11}\\},\\{a^{\prime}_{12},a^{\prime}_{01}\\},e_{2}:=\\{a^{\prime}_{02},c^{\prime}\\}$
must also appear in $\mathbb{G}$. We note that the first two edges differ by
an automorphism of $\mathrm{I}$ so we may treat only one of them explicitly
(note that the proposed necessary set is stable under automorphism of
$\mathrm{I}$). If the edge $e_{1}$ is in $\mathbb{G}$, then the set
$\\{a^{\prime}_{02},a^{\prime}_{11},c^{\prime},a^{\prime}_{01}\\}$ forms a
subgraph isomorphic to $\mathrm{III}_{4}$, which means that either $e_{2}$ or
$e_{3}=\\{a^{\prime}_{11},a^{\prime}_{01}\\}$ appear in $\mathbb{G}$.

Thus the statement that either $e_{1}$ or $e_{2}$ appears in $\mathbb{G}$ can
be reduced to the statement that either $e_{2}$ or $e_{3}$ (the dotted edges)
appear in $\mathbb{G}$. To show the desired result it will be enough to check
that there are two intersection graphs having $\mathrm{I}$ as a subgraph with
the edge $e_{2}$ but not $e_{3}$ and vice versa. To do this, we note that the
following two graphs are interval intersection graphs in $\mathbb{R}$ (with
labels indicating a possible set of intervals with the given intersection
graph):

$(4{,}\,9)$$(6{,}\,7)$$(6{,}\,7)$$(2{,}\,5)$$(1{,}\,3)$$(8{,}\,11)$$(10{,}\,12)$
$(7{,}\,10)$$(5{,}\,8)$$(4{,}\,6)$$(2{,}\,8)$$(1{,}\,3)$$(9{,}\,12)$$(11{,}\,13)$

The other edges belong in the $\subseteq$-minimal necessary set by symmetric
arguments. ∎

######  Lemma 8.13.

The dotted edges in

$v_{4}$$v_{3}$$v_{2}$$v_{1}$$v_{5}$$v_{6}$$v_{7}$

form a necessary set of minimal cardinality for $\mathrm{II}$ in $L^{2}$.

###### Proof.

As before, we recognize that the set $A=\\{v_{2},v_{6},v_{7}\\}$ is asteroidal
in $\mathrm{II}$ and start our analysis with edges that are adjacent to
elements of $A$. We start by noticing that the edges
$\\{v_{2},v_{4}\\},\\{v_{4},v_{6}\\},$ and $\\{v_{1},v_{7}\\}$ must be in any
necessary set as $\mathrm{II}$ becomes an interval intersection graph in
$\mathbb{R}$ when any one of these edges is added. For example, the
collections of intervals given by:

$(2{,}\,8)$$(4{,}\,6)$$(4{,}\,6)$$(5{,}\,10)$$(7{,}\,10)$$(9{,}\,10)$$(1{,}\,3)$
$(3{,}\,8)$$(1{,}\,4)$$(1{,}\,2)$$(1{,}\,10)$$(7{,}\,10)$$(9{,}\,10)$$(5{,}\,6)$

(note that the case of the edge $\\{v_{4},v_{6}\\}$ being added is symmetric
to the case of $\\{v_{2},v_{4}\\}$).

It remains to find any intersection graphs $\mathbb{G}$ with $\mathrm{II}$ as
a subgraph and which fail to contain any of the edges so far found. The
remaining choices for adding an edge incident to an element of $A$ are (up to
symmetry) $e_{1}=\\{v_{2},v_{7}\\}$, $e_{2}=\\{v_{2},v_{5}\\}$, and
$e_{3}=\\{v_{2},v_{6}\\}$. Note that the edge $\\{v_{3},v_{7}\\}$ (or the
symmetric $\\{v_{5},v_{7}\\}$) can be added to $\mathrm{II}$ without changing
the fact that $A$ is asteroidal. Thus another edge incident to an element of
$A$ must be still be added to produce an intersection graph, allowing us to
ignore the edge $\\{v_{3},v_{7}\\}$.

Suppose first that an intersection graph $\mathbb{G}$ extends $\mathrm{II}$
and has the edge $e_{1}$. Then the vertices $\\{v_{1},v_{4},v_{7},v_{2}\\}$
form a cycle of length 4, and so either the edge $\\{v_{1},v_{7}\\}$ or the
edge $\\{v_{2},v_{4}\\}$ belong to $\mathbb{G}$. Both of these edges are
already accounted for, so all intersection graphs with the edge
$\\{v_{2},v_{7}\\}$ are already accounted for.

Now suppose that $\mathbb{G}$ is an intersection graph containing the edge
$\\{v_{2},v_{5}\\}$. But then $\mathbb{G}$ contains the 4-cycle
$v_{2},v_{3},v_{4},v_{5}$ and so must contain either the edge
$\\{v_{3},v_{5}\\}$ or the edge $\\{v_{2},v_{4}\\}$. The latter is already
covered, so adding $\\{v_{3},v_{5}\\}$ to the necessary set will cover all
intersection graphs with the edge $\\{v_{2},v_{5}\\}$ (and, by a symmetric
argument, all intersection graphs with the edge $\\{v_{3},v_{6}\\}$). It
remains to check that the edge $\\{v_{3},v_{5}\\}$ occurs in an intersection
graph not having any of the other edges in the proposed necessary set. The
graph $\mathrm{II}$ with the edges $\\{v_{2},v_{5}\\}$ and $\\{v_{3},v_{5}\\}$
added accomplishes this goal. In particular, the collection of real intervals

$(2{,}\,7)$$(6{,}\,9)$$(8{,}\,10)$$(4{,}\,13)$$(5{,}\,12)$$(11{,}\,14)$$(1{,}\,3)$

has this intersection graph.

Finally suppose that $\mathbb{G}$ is an intersection graph extending
$\mathrm{II}$ and containing the edge $\\{v_{2},v_{6}\\}$. Then the vertices
$\\{v_{2},v_{6},v_{5},v_{4},v_{3}\\}$ form a subgraph isomorphic to
$\mathrm{III}_{5}$ labels occurring in the given order, so one of the edges
$\\{v_{2},v_{5}\\}$, $\\{v_{6},v_{4}\\}$, or $\\{v_{6},v_{3}\\}$ must also be
in $\mathbb{G}$. Any graph containing the first two of these edges has already
been explicitly covered, and any intersection graph with $\\{v_{3},v_{6}\\}$
is covered by an argument symmetric to the argument for graphs containing
$\\{v_{2},v_{5}\\}$. Thus every such $\mathbb{G}$ already has an edge in our
proposed necessary set.

This necessary set is of minimal cardinality because all necessary sets must
contain all but one of the proposed edges ∎

######  Lemma 8.14.

The dotted edges in

$\beta$$\delta$$\gamma$$\alpha$$1$$2$$3$$m$

form a necessary set of minimal cardinality for $\mathrm{IV}_{m}$ in $L^{2}$.

###### Proof.

As before, we start by identifying the set $A=\\{\alpha,\beta,\delta\\}$ as
being asteroidal in $\mathrm{IV}_{m}$. The edges $\\{\beta,\gamma\\}$,
$\\{\gamma,\delta\\}$, and $\\{\alpha,i\\}$ for each $1\leq i\leq m$ must be
in every necessary set for $\mathrm{IV}_{m}$ as $\mathrm{IV}_{m}$ becomes an
intersection graph as soon as any one of the above edges is added. Up to
symmetry, representative examples of these intersection graphs are:

$(3,4)$$(7,8)$$(1,6)$$(1,2)$$(3,6)$$(5,8)$
$(1,2)$$(11,12)$$(3,10)$$(6,7)$$(1,5)$$(4,9)$$(8,12)$

It remains to show that any intersection graph extending $\mathrm{IV}_{m}$ and
not containing any of the aforementioned edges must contain the edge
$\\{1,m\\}$ (and that such intersection graphs do exist).

If $\mathbb{G}$ extends $\mathrm{IV}_{m}$ and $\\{\alpha,\beta\\}$ is in
$\mathbb{G}$ then $\\{\alpha,\beta,1,\gamma\\}$ forms a 4-cycle and so either
$\\{\alpha,1\\}$ is in $\mathbb{G}$ or $\\{\beta,\gamma\\}$ is in
$\mathbb{G}$. However, both of these options are already covered, so
$\\{\alpha,\beta\\}$ cannot be in a $\subseteq$-minimal necessary set. A
symmetric argument gets the same result for the edge $\\{\alpha,\delta\\}$.

Now, in order to avoid $A$ being asteroidal, an intersection graph extending
$\mathrm{IV}_{m}$ and having none of the previously mentioned edges must have
one of the edges $\\{\beta,\delta\\}$, $\\{\beta,m\\}$, or $\\{\delta,1\\}$.
If $\\{\beta,\delta\\}$ is an edge of $\mathbb{G}$ then the vertices
$\\{\beta,\delta,m,\gamma,1\\}$ form a 5-cycle so one of the edges
$\\{1,m\\},\\{\gamma,\delta\\},$ or $\\{\gamma,\beta\\}$ must appear in
$\mathbb{G}$, but we have already ruled out the latter two, so $\\{1,m\\}$
must be in $\mathbb{G}$.

If $\\{\beta,m\\}$ is in $\mathbb{G}$ then the vertices
$\\{\beta,m,\gamma,1\\}$ form a 4-cycle and so either $\\{\beta,\gamma\\}$ is
in $\mathbb{G}$ or $\\{1,m\\}$ is in $\mathbb{G}$. We already know that graphs
with $\\{\beta,\gamma\\}$ are covered, so we must have that $\\{1,m\\}$ is in
$\mathbb{G}$. A symmetric argument for $\\{\delta,1\\}$ also shows that
$\\{1,m\\}$ must be an edge in $\mathbb{G}$.

The result now follows from the fact that the extension of $\mathrm{IV}_{m}$
with the vertices $\\{\beta,\delta\\}\cup\\{1,\dots,m\\}$ inducing a complete
subgraph and otherwise having only the edges from $\mathrm{IV}_{m}$ is an
intersection graph. An example of intervals producing such a graph could be
given by setting all of the vertices labelled $1,\dots,m$ to the same non-
empty interval and also identify the vertices labelled $\beta$ and $\delta$.
These identifications lead to a quotient graph of the proposed graph that is
isomorphic to a graph that is a single irreducible path, which was shown to be
an intersection graph in Lemma 8.11. ∎

######  Lemma 8.15.

The dotted edges in

$\delta$$\epsilon$$\alpha$$\beta$$\gamma$$1$$2$$3$$n$

form the unique necessary set of minimal cardinality for $\mathrm{V}_{n}$ in
$L^{2}$.

###### Proof.

We first note that the set $A=\\{\alpha,\beta,\gamma\\}$ is asteroidal in
$\mathrm{V}_{n}$. We also note that if any one of the edges
$\\{\gamma,\delta\\}$, $\\{\alpha,\epsilon\\}$, or $\\{\beta,i\\}$ for some
$1\leq i\leq n$ are added to $\mathrm{V}_{n}$, then the resulting graph is an
intersection graph, and so every necessary set for $\mathrm{V}_{n}$ contains
the edges $\\{\alpha,\epsilon\\}$, $\\{\gamma,\delta\\}$, and $\\{\beta,i\\}$
for each $1\leq i\leq n$. Representative examples of such graphs are (up to
symmetry):

${(1,8)}$${(1,9)}$${(3,7)}$${(1,2)}$${(10,11)}$${(3,6)}$${(5,9)}$${(7,11)}$
${(1,10)}$${(3,12)}$${(1,2)}$${(6,7)}$${(11,12)}$${(3,5)}$${(4,9)}$${(8,10)}$

Suppose that $\mathbb{G}$ is an intersection graph extending $\mathrm{V}_{n}$
and containing the edge $\\{\alpha,\beta\\}$. Then the vertices
$\\{\alpha,\beta,\epsilon,1\\}$ form a 4-cycle, and so one of the edges
$\\{\beta,1\\}$ or $\\{\alpha,\epsilon\\}$ must be contained in $\mathbb{G}$
as well. However, we already know that both of these edges must be in the
necessary set.

If $\mathbb{G}$ contains the edge $\\{\alpha,\gamma\\}$ then the vertices
$\\{\alpha,\gamma,\epsilon,\delta\\}$ form a 4-cycle, so one of the edges
$\\{\alpha,\epsilon\\}$ or $\\{\gamma,\delta\\}$ must be in $\mathbb{G}$.
Again, we already know that both of these edges must be in every necessary
set.

At this point, no other edges can prevent $A$ from being asteroidal in an
extension, so we have covered all the required cases. ∎

Now that we have computed the minimal necessary sets for each of the minimal
graphs that cannot appear as induced subgraphs of interval intersection
graphs, we can state a theorem for $\mathrm{SOP}$ that is an analogue of
Theorem 8.4.

######  Theorem 8.16.

Let $\mathcal{D}$ be a regular ultrafilter on the infinite cardinal $\lambda$.
The following are equivalent:

  1. (1)

$\mathcal{D}$ is good.

  2. (2)

$\mathcal{D}$ saturates some theory with $\mathrm{SOP}$.

  3. (3)

If $(\mathbb{G}_{\alpha})_{\alpha<\lambda}$ is a sequence of finite graphs
such that
$\mathrm{I},\mathrm{II},\mathrm{III}_{\ell},\mathrm{IV}_{m},\mathrm{V}_{n}\nleq\mathbb{G}_{\alpha}$
for each $\alpha$ and $\ell\geq 4,m\geq 2,$ and $n\geq 1$, then every complete
$\mathbb{H}\subseteq\mathbb{G}_{\alpha}^{\mathcal{D}}=:\mathbb{G}$ with
$|\mathbb{H}|\leq\lambda$ has some internal complete subgraph of $\mathbb{G}$
containing it.

  4. (4)

Every graph-like function $f:\mathcal{P}_{\omega}(\lambda)\to\mathcal{D}$ that
satisfies the conditions given by specializing the subset relation (7.2) to
the necessary sets found for
$\mathbb{H}=\mathrm{I},\mathrm{II},\mathrm{III}_{\ell},\mathrm{IV}_{m},\mathrm{V}_{n}$
in Lemmas 8.11–8.15 has a multiplicative refinement.

###### Proof.

$(1)$$\iff$$(2)$: Follows from e.g. [5, Thm. 2.9].

$(2)$$\iff$$(3)$: A special case of Theorem 7.17.

$(2)$$\iff$$(4)$: Follows from Theorem 7.16 and Lemmas 8.11–8.15. ∎

###  8.3. $\mathrm{SOP}_{n}$ for $n>2$

In this section, our goal is to show that graph-like distributions
corresponding to types whose realizations would fill a cut in an ultrapower of
an infinite linear order (that is, types that characteristically appear in
theories with $\mathrm{SOP}$) also appear as graph-like distributions for
types that appear in theories with $\mathrm{SOP}_{3}$, regardless of whether
they also have $\mathrm{SOP}$. In essence this will show that distributions in
theories with $\mathrm{SOP}_{n}$ for $n\geq 3$ are not only complicated enough
to guarantee that the theory is Keisler maximal but are complicated enough to
mimic the distributions of theories where models contain infinite linear
orders. This argument is a close cousin to the argument made in [5, Thm 2.9]
that theories with $\mathrm{SOP}_{3}$ are Keisler maximal, but is more
elementary.

######  Theorem 8.17.

Suppose that $\mathcal{D}$ is a regular ultrafilter on the infinite cardinal
$\lambda$, $\mathbb{L}=(L,<)$ is a linear order, $\varphi(x;y,z)$ is the
formula $y<x<z$, and
$p(\bar{x})=\\{\varphi(\bar{x};a_{\beta},b_{\beta}):\beta<\lambda\\}$ is a
type of $\mathbb{L}^{\mathcal{D}}$. Then, for every distribution $f$ of
$p(\bar{x})$ and graph-like distribution $\hat{f}$ extending $f$ and every
complete first-order theory $T$ with $\mathrm{SOP}_{3}$, there is a model
$\mathbb{M}\vDash T$ and a graph-like type $q(\bar{y})$ of
$\mathbb{M}^{\mathcal{D}}$ with a distribution $\hat{f}^{\prime}$ satisfying
$\hat{f}=\hat{f}^{\prime}$ (up to bijection between $p(\bar{x})$ and
$q(\bar{y})$).

###### Proof.

Fix representations of $a_{\beta},b_{\beta}$ and a distribution $f$ of
$p(\bar{x})$ given these representations. Since $p$ is graph-like under any
choice of representations in $L$, there is a graph-like distribution $\hat{f}$
extending $f$. Let the distribution graph sequence of $\hat{f}$ be
$(\mathbb{G}_{\alpha})_{\alpha<\lambda}$ where each $\mathbb{G}_{\alpha}$ is a
finite intersection graph (see Lemma 8.6). Let $I(\lambda)$ be the collection
of open intervals $I$ in $\lambda$ with the property that $\min(I)$ is a
successor ordinal. For each $\alpha<\lambda$ let $i_{\alpha}:\lambda\to
I(\lambda)$ be a function such that
$i_{\alpha}\text{\,}\mathord{\upharpoonright}\text{\,}_{\mathbb{G}_{\alpha}}$
is a bijection and $i_{\alpha}(\beta)$ and $i_{\alpha}(\gamma)$ intersect if
and only if $\beta$ and $\gamma$ are neighbors in $\mathbb{G}_{\alpha}$.
Furthermore, we require that no two intervals in the image of $i_{\alpha}$
have endpoints in common. Let the functions $\ell,r:I(\lambda)\to\lambda$ be
the functions defined by

$\ell(I)=\inf(I)\;\text{ and }\;r(I)=\sup(I).$

The function $\ell(I)$ is well-defined by the requirement that $\min(I)$ be a
successor ordinal.

By [5], $T$ has $\mathrm{SOP}_{3}$ if and only if there is a model
$\mathbb{M}\vDash T$ containing an indiscernible sequence
$(\bar{a}_{i})_{i\in\omega}$ and formulas $\theta(\bar{x},\bar{y})$ and
$\eta(\bar{x},\bar{y})$ such that

  1. (1)

the set $\\{\theta(\bar{x},\bar{y}),\eta(\bar{x},\bar{y})\\}$ is
contradictory,

  2. (2)

for some sequence $(b_{j})_{j\in\omega}$ we have

$i\leq j\Rightarrow\theta(\bar{b}_{j},\bar{a}_{i})\;\,\text{ and
}\;\,i>j\Rightarrow\eta(\bar{b}_{j},\bar{a}_{i})$

  3. (3)

for $i<j$, the set
$\\{\theta(\bar{x},\bar{a}_{j}),\eta(\bar{x},\bar{a}_{i})\\}$ is
contradictory.

Using compactness, we may extend the sequences $\bar{a}$ and $\bar{b}$ to
sequences indexed by $\lambda$. Let $(\bar{a}_{i})_{i\in\lambda}$,
$(\bar{b}_{j})_{j\in\lambda}$, $\theta(\bar{x},\bar{y})$, and
$\eta(\bar{x},\bar{y})$ be as above. We then define

$\psi(\bar{x};\bar{y},\bar{z})\equiv\theta(\bar{x},\bar{y})\wedge\eta(\bar{x},\bar{z}).$

Furthermore, define $(\bar{c}_{\beta})_{\beta\in\lambda}$ and
$(\bar{d}_{\beta})_{\beta\in\lambda}$ in $\mathbb{M}^{\mathcal{D}}$ by the
representations

$\bar{c}_{\beta}[\alpha]=\bar{a}_{\ell(i_{\alpha}(\beta))}\;\text{ and
}\;\bar{d}_{\beta}[\alpha]=\bar{a}_{r(i_{\alpha}(\beta))}.$

We claim that the set of formula

$q(\bar{x})=\\{\psi(\bar{x};\bar{c}_{\beta},\bar{d}_{\beta}):\beta<\lambda\\}$

is a graph-like type with distribution $\hat{f}$.

In order to check that $q(\bar{x})$ is graph-like, suppose that
$\Delta\in\mathcal{P}_{\omega}(\lambda)$ is such that $|\Delta|\geq 2$. We
wish to show that

$L_{q}(\Delta)=\bigcap_{\Phi\in[\Delta]^{2}}L_{q}(\Phi),$

where $L_{q}$ is the Łoś map for $q(\bar{x})$. In order to do this, we will
instead show that

(8.1) $L_{q}(\Delta)=\Big{\\{}\alpha\in\lambda:\bigcap
i_{\alpha}[\Delta]\neq\emptyset\Big{\\}},$

which will be enough because of the fact that an intersection of open
intervals is non-empty if and only if the intersection of every pair of
intervals in the collection is non-empty. Suppose that $j\in\bigcap
i_{\alpha}[\Delta]$. Then we have that
$\mathbb{M}\vDash\psi(\bar{b}_{j};\bar{c}_{\beta}[\alpha],\bar{d}_{\beta}[\alpha])$
for each $\beta\in\Delta$ because
$\ell(i_{\alpha}(\beta))<j<r(i_{\alpha}(\beta))$ by the definition of $j$. Now
suppose that $\bigcap i_{\alpha}[\Delta]=\emptyset$. Then there is some
distinct pair $\beta,\gamma\in\Delta$ such that
$r(i_{\alpha}(\beta))\leq\ell(i_{\alpha}(\gamma))$. Furthermore, because the
intervals in the image of $i_{\alpha}$ cannot repeat endpoints, this
inequality is strict. In particular, this means that the formula

$\psi(\bar{x};\bar{c}_{\beta}[\alpha],\bar{d}_{\beta}[\alpha])\wedge\psi(\bar{x};\bar{c}_{\gamma}[\alpha],\bar{d}_{\gamma}[\alpha])$

cannot be satisfied in $\mathbb{M}$.

We check that $\hat{f}$ is a distribution of the type $q(\bar{x})$ with the
given representations. By the fact that $q(\bar{x})$ and $\hat{f}$ are graph-
like, it will be enough to check that ${\hat{f}(\Delta)\subseteq L(\Delta)}$
for all $\Delta\in\mathcal{P}_{\omega}(\lambda)$ with ${|\Delta|\leq 2}$.
However, this follows immediately from Equation (8.1) and the fact that
$\hat{f}$ is a distribution for $p(\bar{x})$. ∎

######  Corollary 8.18.

Theorem 8.17 is true whenever $T$ is replaced with a theory with
$\mathrm{SOP}_{n}$ for $n>3$.

###### Proof.

By [5], $\mathrm{SOP}_{n}\Rightarrow\mathrm{SOP}_{3}$ whenever $n\geq 3$. ∎

Given the difficulty in proving that theories with $\mathrm{SOP}_{2}$ are
maximal in Keisler’s order, we believe that the above theorem is not true for
$T$ with $\mathrm{SOP}_{2}$. We note that if this conjecture is true, then
$\mathrm{SOP}_{3}$ and $\mathrm{SOP}_{2}$ must be different on the level of
theories as has been previously conjectured by Shelah.

## Acknowledgements

Much thanks goes to Ágnes Szendrei for boundless advice given throughout the
writing process and for suggesting looking at distributions related to
$\mathrm{SOP}_{n}$ for $n>2$. This material is based upon work supported by
the National Science Foundation grant no. DMS1500254.

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