A Wowzer-type lower bound for the strong regularity lemma

Subrahmanyam Kalyanasundaram*, Asaf Shapira

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The regularity lemma of Szemerédi 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 et al. ('Efficient testing of large graphs', Combinatorica 20 (2000) 451-476) 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 guaranteed only 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.

Original languageEnglish
Pages (from-to)621-649
Number of pages29
JournalProceedings of the London Mathematical Society
Volume106
Issue number3
DOIs
StatePublished - Mar 2013
Externally publishedYes

Funding

FundersFunder number
National Science FoundationDMS-0901355
Seventh Framework Programme303320
Marie Curie
Israel Science Foundation224/11

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