A Tight Bound for Hyperaph Regularity

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≥ 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.