A Tight Bound for Hyperaph Regularity

Abstract

The hypergraph regularity lemma -- the extension of Szemerédi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle-Rödl-Schacht-Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the $k$-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $k\ge 2$, thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers' famous lower bound for graph regularity.

To appear in GAFA. This is a merged version of arXiv:1804.05511 and arXiv:1804.05513. See arXiv:1804.05511 for a self contained proof of the theorem for the special case of 3-uniform hypergraphs