A Simple Regularization of Hypergraphs

Abstract

We give a simple and natural (probabilistic) construction of hypergraph regularization. It is done just by taking a constant-bounded number of random vertex samplings only one time (thus, iteration-free). It is independent from the definition of quasi-randomness and yields a new elementary proof of a strong hypergraph regularity lemma. Consequently, as an example of its applications, we have a new self-contained proof of Szemer\'edi's classic theorem on arithmetic progressions (1975) as well as its multidimensional extension by Furstenberg-Katznelson (1978).