I'am reading the paper Elementary submodels in infinite combinatorics by Soukup (arXiv link) 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.
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, 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, 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.
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 involves elementary substructures of some kind.
Decent introductions can be found in Discovering modern set theory. II, by Weese and Just, and in Introduction to cardinal arithmetic, by Holz, Steffens, and Weitz. Most likely there are a few other presentations at a comparable level.
The technique is so useful that is employed in many arguments, in a large variety of ways, and I fear I would leave out too many references even if I just tried to list relevant examples covering the key applications.
That said, you definitely want to examine arguments related to negative partition relations, that is, arguments establishing the existence of colorings where large sets not only fail to be monochromatic, but actually take as many colors as possible. There are three papers I suggest you look at in this regard: Consider the following theorem:
Suppose $\lambda$ is an uncountable regular cardinal and there is some non-reflecting stationary subset of $\lambda$. Then $\lambda\not\to[\lambda]^2_\lambda$. That is, there is a function $f:[\lambda]^2\to\lambda$ such that whenever $X$ is a subset of $\lambda$ of size $\lambda$, we have that $f[[X]^2]=\lambda$.
This result was proved by Shelah using Todorcevic's method of walks, and it is an elegant elementary substructure argument. See
Saharon Shelah. Was Sierpiński right? I. Israel J. Math., 62 (3), (1988), 355–380. MR0955139 (89m:03037).
The method of walks itself is central to many modern results in infinitary combinatorics. Several key properties (of the so-called $\rho$-functions) are established by elementary substructure arguments. Todorcevic's introduction, where $\omega_1\not\to[\omega_1]^2_{\omega_1}$ is first established, is highly recommended:
Stevo Todorcevic. Partitioning pairs of countable ordinals. Acta Math., 159 (3-4), (1987), 261–294. MR0908147 (88i:04002).
Todorcevic's result was later extended by Moore, showing that we can in fact ensure the existence of an $f:[\omega_1]^2\to\omega_1$ such that for any uncountable $A,B\subseteq\omega_1$, $f[A\otimes B]=\omega_1$, that is, for any $\xi<\omega_1$ there are $\alpha<\beta$, with $\alpha\in A$ and $\beta\in B$ such that $f(\alpha,\beta)=\xi$. This is a key component of his solution to the $L$-space problem:
Justin T. Moore. A solution to the $L$ space problem. J. Amer. Math. Soc., 19 (3), (2006), 717–736. MR2220104 (2008c:54022).
Even here, the list above is severely outdated: For example, there are fairly recent results of Rinot and Rinot-Todorcevic on the "rectangular square-bracket operation", and Stevo has written a beautiful book on the technique of walks, Walks on ordinals and their characteristics. Also, Eisworth-Shelah have extensions to coloring theorems at successors of singular cardinals.
The method of elementary substructures is also used to establish positive partition relations. A very nice paper, fairly readable and a good way of acquiring some intuition on the technique, is
James E. Baumgartner, András Hajnal, Stevo Todorcevic. Extensions of the Erdős-Rado theorem. In Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), pp. 1–17, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Kluwer Acad. Publ., Dordrecht, 1993. MR1261193 (95c:03111).
A direct continuation of the methods used in this paper, but now to establish polarized relations, is in
James E. Baumgartner, András Hajnal. Polarized partition relations, J. Symbolic Logic, 66 (2), (2001), 811–821. MR1833480 (2002k:03068).
A good survey of these methods can also be seen in the Hajnal-Larson paper for the Handbook.
I expect the above directly addresses your question, but I would be remiss if I did not mention that the technique is also used in many forcing arguments. For example, to establish chain conditions. There is also Todorcevic's technique of forcing with models as side conditions, which recently has been used by Friedman and Neeman, independently, to obtain an essentially different proof of the consistency of $\mathsf{PFA}$ relative to a supercompact cardinal, in the process answering a question of mine and others on the reals of inner models of models of $\mathsf{PFA}$.
Again, it would be futile to even attempt to give a reasonably complete list of suggestions here. But I expect the above should give you a good starting point anyway. Let me add that the technique is particularly useful when studying proper and semiproper forcing notions (this is emphasized in the Martin's maximum paper by Foreman, Magidor, and Shelah, for example).
I still hope one day (soon...) to finish a book Boban Velickovic and I started a long while ago. For now, I have posted on my site a short excerpt showing how the techniques of elementary substructures can be used even at a very basic level.
(A more sophisticated application by Boban and I can be seen in this paper.)
