A Tight Bound for Hypergraph Regularity II

TL;DR

This paper proves that the bounds for hypergraph regularity lemmas grow at an Ackermann-type rate for all uniformities, confirming a long-standing prediction and extending previous results from 3-uniform to all hypergraphs.

Contribution

It extends the known Ackermann-type lower bounds for hypergraph regularity lemma bounds from 3-uniform hypergraphs to all k-uniform hypergraphs, confirming Tao's prediction.

Findings

Ackermann-type bounds are unavoidable for all k ≥ 2 hypergraph regularity lemmas.

The result generalizes previous 3-uniform hypergraph bounds to all uniformities.

Confirms the predicted growth rate of regularity lemma bounds for hypergraphs.

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.