Szemerédi's regularity lemma revisited

Szemerédi's Regularity Lemma and Its Applications in Graph Theory

Szemerédi's Regularity Lemma for Matrices and Sparse Graphs

Szemerédi’s Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.

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.

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.

A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity

The following sharpening of Turán’s theorem is proved. Let Tn,p denote the complete p– partite graph of order n having the maximum number of edges. If G is an n-vertex Kp+1-free graph with e(Tn,p) − t edges then there exists an (at most) p-chromatic subgraph H0 such that e(H0) ≥ e(G)− t. Using this result we present a concise, contemporary proof (i.e., one using Szemerédi’s regularity lemma) fo...