Notes on the Stable Regularity Lemma

Abstract

This is a short expository account of the regularity lemma for stable graphs proved by the authors, with some comments on the model theoretic context, written for a general logical audience.Some years ago, we proved a "stable regularity lemma" showing essentially that Szemerédi's celebrated regularity lemma is much stronger for graphs which do not contain large half-graphs [11, Theorem 5.18], thus characterizing the existence of irregular pairs in Szemerédi's lemma by instability in the sense of model theory.Since that time, it has been a pleasure to see the work which has grown out from this theorem, with various interesting extensions, further developments, and new directions worked out by many different colleagues. Nonetheless, it seems the clear 'picture' of the original proof has not necessarily been widely communicated. Perhaps having a short exposition available may help inspire further interactions and applications.So in these brief expository notes we give a short overview of the proof itself and the model-theoretic ideas behind the proof. Recall that our story begins with:Theorem A (Szemerédi's regularity lemma, 1978). For every > 0 there is N ( ) s.t. every finite graph G may be partitioned into• all but at most m 2 of the pairs (V i , V j ) are -regular.Thanks: Malliaris was partially supported by NSF DMS-1553653 and by a Minerva Research Foundation membership at the IAS. Shelah was partially supported by ERC grant 338821. Both authors thank NSF grant 1362974 (Rutgers) and ERC 338821. This is paper E98 in Shelah's list. These notes benefitted from lectures in the Chicago REU over several summers. We are grateful to E. Bajo, C. Terry, and the anonymous referee for very helpful comments on the manuscript.1 This is a ``preproof'' accepted article for The Bulletin of Symbolic Logic. This version may be subject to change during the production process.

Countable Ramsey

“…However, if P is property of graphons that is closed under taking subgraphons, then a graphon W has a subgraphon satisfying P if and only if it has a positive measure U ⊆ X such that W | U ×U satisfies P . The backward implication is obvious, and the forward implication can be seen easily from Lemma 3.3: if the subgraphon W satisfying P corresponds to a measurable function f of positive integral, then for some > 0, the set The main ingredient to prove this theorem is the following lemma whose main idea can be seen as a graphon analogue of the construction of -good sets in [MS14,MS21], but with = 0 and is much easier for measure theoretic reasons.…”

“…Another important notion in [MS14,MS21] that can be generalized to graphons is that of excellent sets. Let us say that a 0-excellent set in a graphon W is a 0-good set 2 U such that for every 0-good set V we either have almost all edges between U and V or we have almost no edges between U and V in the sense that 1 µ(U × V ) U ×V W (x, y) dµ(x, y) ∈ {0, 1}.…”

“…It was very interesting to us to discover in the course of writing this paper that stability appears in a characteristic sense in some of our main results, despite investigating a priori unrelated questions. Stability is not only a key concept in Shelah's classification theory [She90] but has had recent applications in finite combinatorics via the stable regularity lemma and stable Ramsey's Theorem [MS14] (see also [AFP18,MS21]) and learning theory via Littlestone dimension [ALMM19,BLM20]. Stability has several equivalent definitions in the language of model theory.…”