Regularity Lemmas for Graphs

Dissertation Defense On Algorithmic Hypergraph Regularity

Thomason and Chung, Graham and Wilson were the first to systematically study quasirandom graphs and hypergraphs and showed that several properties of random graphs imply each other in a deterministic sense. In particular, they showed that ε-regularity from Szemerédi’s regularity lemma is equivalent to their concepts. Over recent years several hypergraph regularity lemmas were established. In th...

Hypergraph regularity and quasi-randomness

Thomason and Chung, Graham, and Wilson were the first to systematically study quasi-random graphs and hypergraphs, and proved that several properties of random graphs imply each other in a deterministic sense. Their concepts of quasi-randomness match the notion of ε-regularity from the earlier Szemerédi regularity lemma. In contrast, there exists no “natural” hypergraph regularity lemma matchin...

Regular Partitions of Hypergraphs: Regularity Lemmas

Szemerédi’s regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and authors and obtain a stronger and more “user friendly” regularity lemma for hypergraphs.

Quasirandomness, Counting and Regularity for 3-Uniform Hypergraphs

The main results of this paper are regularity and counting lemmas for 3uniform hypergraphs. A combination of these two results gives a new proof of a theorem of Frankl and Rödl, of which Szemerédi’s theorem for arithmetic progressions of length 4 is a notable consequence. Frankl and Rödl also prove regularity and counting lemmas, but the proofs here, and even the statements, are significantly d...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs