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arXiv · 1804.05513

A Tight Bound for Hypergraph Regularity II

Guy Moshkovitz·Asaf Shapira
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Abstract

The hypergraph regularity lemma -- the extension of Szemer\'edi'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-R\"odl-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. In a recent paper we have shown that these bounds are unavoidable for $3$-uniform hypergraphs. In this paper we extend this result by showing that such Ackermann-type bounds are unavoidable for every $k \ge 2$, thus confirming a prediction of Tao.

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Guy Moshkovitz, Asaf Shapira. 2018-04-16. A Tight Bound for Hypergraph Regularity II. https://arxiv.org/abs/1804.05513

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