Szemerédi's Regularity Lemma for Matrices and Sparse Graphs

Szemerédi's Regularity Lemma via Martingales

We prove a variant of the abstract probabilistic version of Szemerédi’s regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in Lp for any p > 1. Our approach is based on martingale difference sequences.

Extremal results in sparse pseudorandom graphs

Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity lemma us...

Szemerédi's Regularity Lemma and Its Applications in Graph Theory

A tight lower bound for Szemerédi's regularity lemma

Addressing a question of Gowers, we determine the order of the tower height for the partition size in a version of Szemerédi’s regularity lemma.

The Regularity Lemma of Szemerédi for Sparse Graphs

In this note we present a new version of the well-known lemma of Szemerédi [17] concerning regular partitions of graphs. Our result deals with subgraphs of pseudo-random graphs, and hence may be used to partition sparse graphs that do no contain dense subgraphs.