Abstract

Advancing the sparse regularity method, we prove one‐sided and two‐sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox, and Zhao. These inheritance lemmas also imply improved H‐counting lemmas for subgraphs of bijumbled graphs, for some H.

Highlights

  • Over the past 40 years, the regularity method has developed into a powerful tool in discrete mathematics, with applications in combinatorial geometry, additive number theory and theoretical computer science

  • The regularity lemma states that each graph can be partitioned into a bounded number of regular pairs

  • The main obstacle here is that in sparse graphs it is no longer true that neighborhoods of vertices in regular pairs are typically large and trivially induce regular pairs—they are of size pn ≪ εn

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Summary

INTRODUCTION

Over the past 40 years, the regularity method has developed into a powerful tool in discrete mathematics, with applications in combinatorial geometry, additive number theory and theoretical computer science (see [12, 15, 18, 21] for surveys). After partial results were obtained in [16], Conlon, Fox, and Zhao [7] recently proved a general counting lemma for subgraphs of bijumbled graphs. The main obstacle here is that in sparse graphs it is no longer true that neighborhoods of vertices in regular pairs are typically large and trivially induce regular pairs—they are of size pn ≪ εn. For all but at most at most ε′|X| vertices x of X, the pair NΓ(x) ∩ Y, Z is (ε′, d, p)-regular in G Comparing this result with the analog by Conlon, Fox, and Zhao in [7, Proposition 5.1], we need Γ to be a factor Our proofs use the counting lemma for C4 of Conlon, Fox, and Zhao [7] as a fundamental ingredient

Applications
Optimality
PROOF OVERVIEW
Bijumbledness
Sparse regularity
Cauchy-Schwarz
ONE-SIDED INHERITANCE
TWO-SIDED INHERITANCE
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