Efficient Removal without Efficient Regularity

TL;DR

This paper introduces a new approach to the induced $C_4$ removal lemma, achieving the first exponential bound without relying on the regularity lemma, marking significant progress in extremal combinatorics.

Contribution

It develops a novel method to obtain an exponential bound for the induced $C_4$ removal lemma without using the regularity lemma, overcoming a longstanding barrier.

Findings

First exponential bound for induced $C_4$ removal lemma

Removes reliance on regularity lemma for this problem

Progress on a problem posed by Alon in 2001

Abstract

Obtaining an efficient bound for the triangle removal lemma is one of the most outstanding open problems of extremal combinatorics. Perhaps the main bottleneck for achieving this goal is that triangle-free graphs can be highly unstructured. For example, triangle-free graphs might have only regular partitions (in the sense of Szemer\'edi) of tower-type size. And indeed, essentially all the graph properties ${\cal P}$ for which removal lemmas with reasonable bounds were obtained, are such that every graph satisfying ${\cal P}$ has a small regular partition. So in some sense, a barrier for obtaining an efficient removal lemma for property ${\cal P}$ was having an efficient regularity lemma for graphs satisfying ${\cal P}$. In this paper we consider the property of being induced $C_4$-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for the induced $C_4$ removal lemma. We thus obtain the first efficient removal lemma that does not rely on an efficient version of the regularity lemma. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox.