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arXiv · 1107.4896

A Wowzer Type Lower Bound for the Strong Regularity Lemma

Subrahmanyam Kalyanasundaram·Asaf Shapira
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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.

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Subrahmanyam Kalyanasundaram, Asaf Shapira. 2012-06-17. A Wowzer Type Lower Bound for the Strong Regularity Lemma. https://doi.org/10.1112/plms%2Fpds045

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