Persistence and Regularity in Unstable Model Theory, Berkeley, 2009.
PDF,
overview.
Contemporary Mathematics Volume 690, 2017.
PDF
Simplicity: ideals of practice in mathematics and the arts. 51–58, Math. Cult. Arts, Springer, Cham, 2017.
link
Contemporary Mathematics volume 752, 2020, pps. 121-152.
arxiv.
Even though the consequences of stability in the finite have been much studied in the literature, the countable setting seems a priori quite different, namely, in the countably infinite the notion of largeness based on cardinality alone does not reveal any structure as Ramsey's Theorem already provides a countably infinite uniform set in general. In this paper, we show that the natural notion of largeness given by upper density reveals that these phenomena meet in the countable: a countable graph has an almost clique or anti-clique of positive upper density if and only if it has a positive upper density almost stable set. Moreover, this result also extends naturally to countable models of a universal theory in a finite relational language. Our methods explore a connection with the notion of convergence in the theory of limits of dense combinatorial objects, introducing and studying a natural approximate version of the Erdős-Hajnal property that allows for a negligible error in the edges (in general, predicates) but requires linear-sized uniform sets in convergent sequences of models (this is much stronger than what stable regularity can provide as the error is required to go to zero). Finally, surprisingly, we completely characterize all hereditary classes of finite graphs that have this approximate Erdős-Hajnal property. The proof highlights both differences and similarities with the original conjecture.
The first result of the paper is a complete characterization of the hereditary families of graphs that are strongly persistent as precisely those that are closed under substitutions. In fact, we prove an analogous characterization for the natural extension of these properties to structures in finite relational languages. We call graphons (or more generally theons) with the self-similarity property above weakly random. Weak randomness can be seen as a weakening of graph quasirandomness: while the latter requires densities of any finite graph to be the same in any subgraphon, the former only requires the set of finite graphs with positive density to be invariant across subgraphons. A hereditary family F is said to have the weakly random Erdős--Hajnal property (WR) if every graphon that is a limit of graphs in F has a weakly random subgraphon; this includes as a special case the approximate Erdős--Hajnal property (AEHP), where the subgraphon is required to be an almost clique or almost anti-clique. Among families of graphs that are closed under substitutions, we completely characterize the families that belong to WR as those with "few" prime graphs. The classes AEHP and WR can be seen as measuring the complexity of hereditary families in terms of structural variation in their limit objects (here specifically in terms of unavoidable sub-objects), and this work can be seen as initiating a classification theory for hereditary families of finite structures.
.
| space |