Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets
Abstract
For every fixed graph H and every fixed 0 < α < 1, we show that if a graph G has the property that all subsets of size α 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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