A Wowzer Type Lower Bound for the Strong Regularity Lemma

TL;DR

This paper proves that for certain graphs, any regular partition with controlled quasi-randomness must have a number of parts at least as large as a Wowzer-type function, establishing a lower bound on the complexity of such partitions.

Contribution

It demonstrates a lower bound matching the Wowzer-type upper bound for the number of parts in strong regularity lemma partitions, showing the bound is tight.

Findings

Constructed a graph requiring Wowzer-type number of parts for controlled regular partitions

Established that the Wowzer bound is unavoidable for certain graphs

Confirmed the optimality of previous upper bounds on regularity partitions

Abstract

The regularity lemma of Szemeredi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. In some applications however, one would like to have a strong control on how quasi-random these bipartite graphs are. Alon, Fischer, Krivelevich and Szegedy obtained a powerful variant of the regularity lemma, which allows one to have an arbitrary control on this measure of quasi-randomness. However, their proof only guaranteed to produce a partition where the number of parts is given by the Wowzer function, which is the iterated version of the Tower function. We show here that a bound of this type is unavoidable by constructing a graph H, with the property that even if one wants a very mild control on the quasi-randomness of a regular partition, then any such partition of H must have a number of parts given by a Wowzer-type function.