Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph can be decomposed into relatively few 'nearly regular' bipartite subgraphs. In this note the author generalizes this result to arbitrary graphs, at the same time strengthening and simplifying the original bipartite result.

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… Expand

The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.Expand

Szemeredi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some further… Expand

We survey some of the recent results related to the study of sparse graphs using the nowhere dense - somewhere dense dichotomy. Particularly we extend known results related to property testing,… Expand

B bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi are considered and it is shown that they must satisfy a certain local pseudorandom property.Expand