TY - JOUR

T1 - A Wowzer-type lower bound for the strong regularity lemma

AU - Kalyanasundaram, Subrahmanyam

AU - Shapira, Asaf

N1 - Funding Information:
The first author has done this work while being a student in Department of Computer Science and Engineering, Indian Institute of Technology, Hyderabad 502205, India. The second author was supported in part by NSF Grant DMS-0901355, ISF Grant 224/11 and a Marie-Curie CIG Grant 303320.

PY - 2013/3

Y1 - 2013/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84876528168&partnerID=8YFLogxK

U2 - 10.1112/plms/pds045

DO - 10.1112/plms/pds045

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AN - SCOPUS:84876528168

SN - 0024-6115

VL - 106

SP - 621

EP - 649

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 3

ER -