# A Tight Bound for Hyperaph Regularity

## TL;DR

This paper establishes that the bounds for hypergraph regularity partitions are inherently Ackermannian in size, confirming a long-standing prediction and demonstrating the optimality of existing bounds.

## Contribution

The paper proves that Ackermann-type bounds are unavoidable for hypergraph regularity lemmas, extending Gowers' lower bounds from graphs to hypergraphs.

## Key findings

- Ackermann-type bounds are necessary for hypergraph regularity.
- The bounds are tight and cannot be improved to simpler functions.
- This confirms Tao's prediction about the complexity of hypergraph regularity.

## 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. 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.