A new proof of the graph removal lemma

TL;DR

This paper presents a new proof of the graph removal lemma that avoids Szemerédi's regularity lemma, resulting in improved bounds and extending to directed and multicolored cases, answering longstanding questions.

Contribution

It introduces a novel proof technique for the graph removal lemma that yields better bounds and applies to directed and multicolored variants, bypassing traditional methods.

Findings

Avoids Szemerédi's regularity lemma

Provides improved bounds for the lemma

Extends results to directed and multicolored graphs

Abstract

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n^h) copies of H can be made H-free by removing o(n^2) edges. We give a new proof which avoids Szemer\'edi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.