Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets

Details Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets

2008-04-04

arxiv.org/abs/0804.0753v1

Mathematics

Combinatorics

7 pages

Scientific paper

For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph $G(n,p)$ then $G$ behaves like the random graph $G(n,p)$; that is, it is $p$-quasi-random in the sense of Chung, Graham, and Wilson. This solves a conjecture raised by Shapira and solves in a strong sense an open problem of Simonovits and S\'os.

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