A Tight Bound for Hyperaph Regularity

Guy Moshkovitz, Asaf Shapira*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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≥ 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.

Original languageEnglish
Pages (from-to)1531-1578
Number of pages48
JournalGeometric and Functional Analysis
Volume29
Issue number5
DOIs
StatePublished - 1 Oct 2019

Funding

FundersFunder number
Horizon 2020 Framework Programme633509

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