I'am reading the paper *Elementary submodels in infinite combinatorics* by Soukup ([arXiv link](https://arxiv.org/abs/1007.4309)) and there are a lot of proofs using elementary submodels, such as the proof of $\Delta$-system lemma and partitions theorems. However, I don't take the intuition and I would like more examples of the applications of elementary submodels. Anyone knows goods references for it in infinite combinatorics, specially in partition theory?


Thanks.



Elementary submodels in partitions theorems
Complementing Andres's excellent answer, let me simply try to help build your intuition for elementary submodels.


The basic situation is just like the familiar fact that if you
have finitely many group elements $g\_0,\ldots,g\_n$ in a large
group $G$, then they generate a countable subgroup of $G$. One
simply starts with the finite set, and applies the group operation and inverses
in all possible ways to what one has generated so far, arriving in countably many steps in a countable set that is closed under inverses and any
further applications of the group operation.


More generally, and for essentially similar reasons, if one has
countably many functions $f:N^{k}\to N$ on a set $N$, then any countable subset $A\subset N$ is included
in a countable set $M\subset N$ that is closed under all the
operations $f$. One simply applies the functions in all possible
ways to what one has generated so far, and there are only
countably many ways to do this.


In particular, consider the case where $N$ is a structure of some
kind, in a countable first order language. Perhaps we have a
formula $\varphi$, such that for some $\vec a$ in $N$, there is a
witness element $x$ for which $\varphi(x,\vec a)$. In this case,
let $f\_\varphi(\vec a)=x$ be a function that selects such a
witness when there is one. Such a function $f\_\varphi$ is called a
*Skolem* function for $\varphi$, a function that selects
existential witnesses when they exist. The observation of the
previous paragraph shows that any countable set $A\subset N$ can
be extended to a countable set $M\subset N$ that is closed under
$f\_\varphi$ for every $\varphi$. But notice that this means that
whenever you have some $\vec a$ in $M$, and have some property
$\varphi(x,\vec b)$ with a witness $x$ in $N$, then you can find
such an $x$ already inside the small set $M$. This feature is
called the [Tarski-Vaught property](http://en.wikipedia.org/wiki/Tarski%E2%80%93Vaught_test), and it is equivalent to
saying that $M$ is an *elementary substructure* of $N$, written
$M\prec N$. Closure under witnesses in this way implies that $M$
and $N$ satisfy exactly the same assertions about objects in $M$
that are expressible in the language of $N$. Thus, we've
essentially proved the [Downward
Löwenheim-Skolem theorem](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem), which says that every first-order
structure in a countable language has a countable elementary
substructure.


The basic theory of elementary substructures is covered in any
introductory logic text.


In the example of the Soukop paper to which you link, the author
uses as $N$ the entire universe $V$ of all sets. In this case,
there are certain meta-mathematical difficulties with getting a
fully elementary substructure, and so they make themselves happy
with a substructure $M$ that is elementary with respect to some
fixed finite collection $\Sigma$ of first-order properties. Thus,
one has $M\prec\_\Sigma V$, which means that $M$ agrees with $V$ on
the truth of any property in $\Sigma$. Such a way of proceeding is
an essential application of the [Reflection theorem](http://en.wikipedia.org/wiki/Reflection_theorem).


The examples of that paper are quite nice. For example, in theorem
4.1 there, he proves the classic $\Delta$-system lemma via
elementary substructures: if $\cal{A}$ is an uncountable
collection of finite sets, then it contains an uncountable
$\Delta$-system. To see this, find a countable elementary
substructure $M\prec\_\Sigma V$, using a suitably large collection
of formulas $\Sigma$, with ${\cal A}\in M$, with $M$ closed under
finite sequences. Let $D=M\cap A$ for some $A\in {\cal A}-M$. In
$V$, there is a maximal $\Delta$-system ${\cal B}\subset{\cal A}$
with root $D$. By elementarity, there is such a $\cal{B}$ inside
$M$. But now $\cal{B}$ cannot be countable, for then it could be
enumerated in $V$ and $M$ would have to have such an enumeration,
and hence also have every element of $\cal{B}$, so ${\cal B}\subset
M$. But in this case, ${\cal B}\cup\{A\}$ would be a strictly
larger subfamily of $\cal{A}$ that is a $\Delta$-system with root
$D$, contrary to the maximality of $\cal{B}$. So it is
uncountable, and we are done. QED


That way of proving the $\Delta$-system lemma is very different
from the other proofs with which I am familiar, and it seems to be
a powerful method.


The method of elementary substructures is used pervasively in set
theory, and there are numerous other applications. For example,
almost any use of [proper forcing](http://en.wikipedia.org/wiki/Proper_forcing_axiom) involves elementary
substructures of some kind.


