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arXiv · 1907.07639

A Tight Bound for Hyperaph Regularity

Guy Moshkovitz·Asaf Shapira
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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.

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Guy Moshkovitz, Asaf Shapira. 2019-07-17. A Tight Bound for Hyperaph Regularity. https://arxiv.org/abs/1907.07639

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