Frankl-Rödl type theorems for codes and permutations

Iterated combinatorial density theorems

Induced restricted Ramsey theorems for spaces

The induced restricted versions of the vector space Ramsey theorem and of the Graham-Rothschild parameter set theorem are proved.

Quantitative theorems for regular systems of equations

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...

Counting Small Cliques in 3-uniform Hypergraphs

Many applications of Szemerédi’s Regularity Lemma for graphs are based on the following counting result. If G is an s-partite graph with partition V (G) = ⋃s i=1 Vi, |Vi| = m for all i ∈ [s], and all pairs (Vi, Vj ), 1 i < j s, are -regular of density d, then G contains (1± f( ))d s 2 ms cliques Ks, provided < (d), where f( ) tends to 0 as tends to 0. Guided by the regularity lemma for 3-unifor...