A Wowzer-type Lower bound for the Strong Regularity Lemma


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 the number of parts in any such partition of $H$ must be given by a Wowzer-type function.

Proceedings of the London Mathematical Society, Volume 106, Issue 3, pages 621–649