A Simple Regularization of Hypergraphs

TL;DR

This paper introduces a simple, probabilistic hypergraph regularization method using a single random sampling, providing an elementary proof of a strong hypergraph regularity lemma and applications to Szemerédi's theorem and its multidimensional extension.

Contribution

It presents a novel, iteration-free hypergraph regularization technique based on random sampling, simplifying proofs of key combinatorial theorems.

Findings

Provides a new elementary proof of the hypergraph regularity lemma.

Offers a self-contained proof of Szemerédi's theorem on arithmetic progressions.

Demonstrates the method's application to multidimensional extensions of Szemerédi's theorem.

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).