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Hereditary quasirandomness without regularity

Published online by Cambridge University Press:  26 January 2017

DAVID CONLON ,
JACOB FOX  and
BENNY SUDAKOV
DAVID CONLON
Affiliation:
Mathematical Institute, Oxford OX2 6GG, United Kingdom. e-mail: david.conlon@maths.ox.ac.uk
JACOB FOX
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail: jacobfox@stanford.edu
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland. e-mail: benjamin.sudakov@math.ethz.ch
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Abstract

A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 385 - 399
Copyright
Copyright © Cambridge Philosophical Society 2017 

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