We give a simple and natural construction of hypergraph regularization. It yields a short proof of a hypergraph regularity lemma. Consequently, as an example of its applications, we have a short self-contained proof of Szemerédi’s classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by Furstenberg-Katznelson (1978).

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...

The central concept in Szemerédi’s powerful regularity lemma is the so-called ε-regular pair. A useful statement of Alon et al. essentially equates the notion of an ε-regular pair with degree uniformity of vertices and pairs of vertices. The known proof of this characterization uses a clever matrix argument. This paper gives a simple proof of the characterization without appealing to the matrix...

Let P : F × F → F be a polynomial of bounded degree over a finite field F of large characteristic. In this paper we establish the following dichotomy: either P is a moderate asymmetric expander in the sense that |P (A,B)| |F| whenever A,B ⊂ F are such that |A||B| ≥ C|F|2−1/8 for a sufficiently large C, or else P takes the form P (x, y) = Q(F (x) + G(y)) or P (x, y) = Q(F (x)G(y)) for some polyn...

We present alternative proofs of density versions of some combinatorial partition theorems originally obtained by H. Furstenberg and Y. Katznelson. These proofs are based on an extremal hypergraph result which was recently independently obtained by W. T. Gowers and B. Nagle, V. Rödl, M. Schacht, J. Skokan by extending Szemerédi’s Regularity Lemma to hypergraphs.

Szemerédi’s Regularity Lemma [18] tells us that every graph can be partitioned into a constant number of sets of vertices in such a way that for most of the pairs of sets in the partition, the bipartite graph of edges between them has many of the properties that one would expect in a random bipartite graph with the same expected edge density. Szemerédi originally used his lemma to prove his cel...

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.

A triangle in a triple system is a collection of three edges isomorphic to {123, 124, 345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all trian...

It is proved that applying sufficient regularity conditions to the interval matrix $[A-|B|,A + |B|]$, we can create a new unique solvability condition for the absolute value equation $Ax + B|x|=b$, since regularity of interval matrices implies unique solvability of their corresponding absolute value equation. This condition is formulated in terms of positive deniteness of a certain point matrix...