On Generalized Regularity

TL;DR

This paper investigates generalized regularity lemmas in graph theory, showing that for all such lemmas, good upper bounds for their quantitative estimates cannot be achieved, highlighting fundamental limitations.

Contribution

It proves that no generalized regularity lemma can have favorable upper bounds for its quantitative estimates, extending previous results to a broad class of regularity lemmas.

Findings

No good upper bounds exist for the estimates of any generalized regularity lemma.

The result generalizes Fan Chung's clustering graph regularity lemma.

Highlights inherent limitations in the applicability of generalized regularity lemmas.

Abstract

Szemeredi's regularity lemma is one instance in a family of regularity lemmas, replacing the definition of density of a graph by a more general coefficient. Recently, Fan Chung proved another instance, a regularity lemma for clustering graphs, and asked whether good upper bounds could be derived for the quantitative estimates it supplies. We answer this question in the negative, for every generalized regularity lemma.