Countable Ramsey

Abstract

The celebrated Erdős-Hajnal Conjecture says that in any proper hereditary class of finite graphs we are guaranteed to have a clique or anti-clique of size n c , which is a much better bound than the logarithmic size that is provided by Ramsey's Theorem in general. On the other hand, in uncountable

Weak randomness in graphons and theons

“…We prove this result first for graphs (Theorem 4.4) and then a suitable generalization of it for structures in arbitrary finite relational languages (Theorem 7.9) after developing suitable extensions of the relevant concepts. The appearance of substitution in this characterization, and of the related notion of primality in what follows, is not completely unexpected as both the Erdős-Hajnal property and its approximate version behave very well under substitution [APS01,CM22].…”

“…Recall that the Erdős-Hajnal Conjecture [EH89] says that for any proper hereditary class of graphs, there exists a constant c > 0 such that any graph of size n in this class either has a clique or an anti-clique of size n c . In [CM22], we studied a natural variant of this question in the presence of convergence, which we called the approximate Erdős-Hajnal property (AEHP), in which we allow for a negligible amount of non-edges in the almost clique or a negligible amount of edges in the almost anti-clique, but require it to be linear-sized. The framework of AEHP naturally lends itself to analysis via limit theory, i.e., graphons [LS06] in the graph case, or more generally, flag algebras [Raz07] and theons [CR20b] in the case of universal theories in finite relational languages.…”

“…Surprisingly, hereditary classes of graphs with AEHP can be characterized as precisely those that avoid containing the aforementioned family F C 4 , see [CM22,Theorem 8.10]. In what follows, it will be more convenient to think about hereditary classes of graphs as the models of a particular universal first-order theory T of graphs, so a graphon of T is simply a limit of finite models of T .…”

“…This shift in language supports the model theoretic perspective of studying the theory T (i.e., a hereditary class of graphs) by studying the variation in the class of its infinite models (i.e., its graphons). In this language, a universal theory T of graphs has AEHP if every graphon of T has a (large) trivial subgraphon, i.e., an almost clique or an almost anti-clique, see [CM22,§7] and Definition 2.1.…”

“…Let us point out that although [CM22] provides a good motivation for the current work, it is not a pre-requisite for the current paper and we do not rely on any of the results of [CM22] for our study of weak randomness and the class WR, except for a straightforward characterization of subgraphons and sub-objects [CM22, Lemmas 3.3 and 5.8] (see also Section 2 below). To read the current paper, it will be useful to have some familiarity with the theories of graphons and theons, but we repeat the relevant definitions and results in Section 2 to set the notation.…”