A Tight Bound for Hypergraph Regularity I

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 the above type was Gowers' famous lower bound for graph regularity. In this paper we describe the key new ideas which enable us to overcome several barriers which stood in the way of establishing such bounds for hypergraphs of higher uniformity. One of them is a tight bound for a new (very weak) version of the {\em graph} regularity lemma. Using this bound, we prove a lower bound for any regularity lemma of $3$-uniform hypergraphs that satisfies certain mild conditions. We then show how to use this result in order to prove a tight bound for the hypergraph regularity lemmas of Gowers and Frankl--R\"odl. We will obtain similar results for hypergraphs of arbitrary uniformity $k\geq 2$ in a subsequent paper.