Small subsets inherit sparse ε-regularity
Journal of Combinatorial Theory Series B
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We prove the existence of many complete graphs in almost allsufficiently dense partitions obtained by an application ofSzemerdi's regularity lemma. More precisely, we consider the numberof complete graphs Kℓ on ℓ verticesin ℓ-partite graphs where each partition class consists ofn vertices and there is an ε-regular graph onm edges between any two partition classes. We show that forall β 0, at most a βm-fraction ofall such graphs contain a little less than the expected number ofcopies of Kℓ provided ε issufficiently small, m n2-1/(ℓ-1), and n is sufficientlylarge. This result is a counting version of a restricted version ofa conjecture (Kohayakawa, Luczak, and Rödl, Combinatorica 17(1997), 173213), and it is well known that this result impliesseveral results for random graphs. © 2007 Wiley Periodicals,Inc. Random Struct. Alg., 2007