Compactness theorem for monster models I am currently reading "A Guide To NIP Theories" by Pierre Simon and I am a bit confused by his use of the compactness theorem. Maybe this stems from the use of a monster model $\mathcal{U}$ ($\bar \kappa$-saturated and homgeneous) which I am unfortunately not familiar with. The situation sets up as follows:
We have a partitioned formula $\phi(x;y)$. Some set $A\subset \mathcal{U}$ of $|x|$-tuples is said to be shattered by $\phi$, if for every $I\subset A$ we can find some $|y|$-tuple $b_I$ s.t.
\begin{align}
\mathcal{U}\models \phi(a,b_I) \iff a\in I, \, \text{for all}\,  a\in A.
\end{align}
The first thing Simon claims is, that by compactness this is the same as saying that every finite subset of $A$ is shattered by $\phi$. Why is that? The set of $L(A)$-formulas
\begin{align}
p((y_I)_{I\subset A}) = \bigcup_{I\subset A} \{\phi(a;y_I)\mid a \in I\}\cup \{\neg \phi(a;y_I)\mid a \in A\setminus I\}
\end{align} is satisfiable iff $A$ is shattered by $\phi$. It is an $A$-type if all finite subsets of $A$ are shattered by $\phi$, so if $|A|< \bar \kappa$ the claim would follow from saturatedness of $\mathcal{U}$. Does he implicitely assume $A$ to be small enough?
Another consequence he draws from the compactness theorem is: If a formula $\phi(x;y)$ shatters finite subsets of $\mathcal{U}$ of arbitrary size then it shatters some infinite subset of $\mathcal{U}$. I'm not entirely sure how that works. I see that compactness guarantees the existence of some model $\mathcal{U}'$ of $\operatorname{Th}(\mathcal{U})$ (that is sufficiently small) and some infinite subset $A\subset \mathcal{U}'$ that is shattered by $\phi$. Does the "monsterness" of $\mathcal{U}$ ensure that the same holds for $\mathcal{U}$ itself?
 A: The answer to your first question is just a matter of notational convention! Whenever Simon refers to a subset or "parameter set" $A\subset\mathcal{U}$, using the strict inclusion symbol $\subset$ instead of $\subseteq$ and letters like $A,B,C$, he implicitly means that this set is small, ie within the saturation of $\mathcal{U}$. This is common notational practice when working in a monster model.
Your second question is resolved by the following fact:
Fact: Let $\kappa$ be a cardinal, and let $M$ be a $\kappa$-saturated first-order structure. Moreover let $A\subset M$ be a subset of size $<\kappa$ and let $p(x_{\alpha}:\alpha\in\lambda)$ be a set of formulas with parameters in $A$ and variables ranging over the variables $(x_\alpha)_{\alpha\in\lambda}$ for some $\lambda\leqslant\kappa$. If $p$ is finitely satisfiable in $M$ then $p$ is satisfiable in $M$. $\blacksquare$
This is a nice exercise, and can be proved by induction on $\lambda$, where the base case of $\lambda=1$ is provided by $\kappa$-saturation; let me know if you have any trouble with it! We would call $p$ a (partial) $\lambda$-type of $M$ over $A$, and we use the usual notation $S^M_\lambda(A)$ to denote the set of all complete $\lambda$-types over $A$.
Anyway, your question is resolved by the case $\lambda=2^{\aleph_0}$, since you can write a partial type in the variables $(x_i)_{i\in\omega}\cup(y_S)_{S\subseteq\omega}$ expressing that $\phi(x_i;y_S)$ holds if and only if $i\in S$ for all $i\in\omega,S\subseteq\omega$. Since $\phi(x;y)$ shatters arbitrarily large finite sets, this partial type is finitely satisfiable in $\mathcal{U}$, and thus by the fact above is satisfiable in $\mathcal{U}$. The realizations of the $x_i$ then give an infinite set shattered by $\phi(x;y)$, as needed.
A: 
Does he implicitly assume  to be small enough?

YES! Usually capital letters A,B,C denoted small subsets of $\scr U$ and  $M,N$ small elementary substructures of $\scr U$.

Another consequence he draws from the compactness theorem is: If a formula (;) shatters finite subsets of $\scr U$ of arbitrary size then it shatters some infinite subset of $\scr U$.

For $n\in\omega$ and $A⊆\omega$ consider the type
$$
p_{n,A}(x_1,\dots,x_n;y_A) = \bigwedge_{i\in A∩n}\phi(x_i;y_A)\ \wedge\  \bigwedge_{i\in n\smallsetminus A}\neg \phi(x_i;y_A)
$$
Let $q((x_i)_{i\in\omega};\ (y_A)_{A\subseteq\omega})$ be the union of all these types.
If $(a_i)$ realizes $\exists\,(y_A)\  q((x_i);\ (y_A))$ then the set $\{a_i:i\in\omega\}$ is shattered. N.B. You can replace $\omega$ with any small cardinal.
