The stable regularity lemma revisited

TL;DR

This paper presents a simplified proof of a regularity lemma for stable definable bipartite graphs with respect to arbitrary Keisler measures, extending previous results to a more general setting.

Contribution

It provides a quick proof of a regularity lemma for stable graphs with respect to Keisler measures, generalizing earlier pseudofinite cases.

Findings

Proves a regularity lemma for stable definable graphs with arbitrary Keisler measures.

Recovers the stable regularity theorem of Malliaris and Shelah as a special case.

Uses local stability theory for a concise proof.

Abstract

We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and the formula R(x,y) is stable. The proof is rather quick and uses local stability theory. The special case where (V,W,R) is pseudofinite, mu, nu are the counting measures and M is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of Malliaris-Shelah (Transactions AMS, 366, 2014, 1551-1585), though without explicit bounds or equitability.