A Sparse Regular Approximation Lemma

TL;DR

This paper introduces the sparse regular approximation lemma (SRAL), a variant of Szemerédi's regularity lemma, which achieves better quantitative bounds and suffices for key applications like the graph removal and hypergraph regularity lemmas.

Contribution

The paper presents the SRAL, providing improved bounds and demonstrating its effectiveness for fundamental combinatorial lemmas, with matching upper and lower bounds on its tower height.

Findings

SRAL achieves tower bounds of height O(log 1/p)

Reproves Fox's upper bound for the graph removal lemma

Establishes a matching lower bound of tower height Ω(log 1/p)

Abstract

We introduce a new variant of Szemer\'edi's regularity lemma which we call the "sparse regular approximation lemma" (SRAL). The input to this lemma is a graph $G$ of edge density $p$ and parameters $\epsilon, \delta$, where we think of $\delta$ as a constant. The goal is to construct an $\epsilon$-regular partition of $G$ while having the freedom to add/remove up to $\delta |E(G)|$ edges. As we show here, this weaker variant of the regularity lemma already suffices for proving the graph removal lemma and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height $O(\log 1/p)$. This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height $\Omega(\log 1/p)$ is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.