Abstract
Let uk(G,p) be the maximum over all k-vertex graphs F of by how much the number of induced copies of F in G differs from its expectation in the binomial random graph with the same number of vertices as G and with edge probability p. This may be viewed as a measure of how close G is to being p-quasirandom. For a positive integer n and 0<p<1, let D(n,p) be the distance from pn2 to the nearest integer. Our main result is that, for fixed k≥4 and for n large, the minimum of uk(G,p) over n-vertex graphs has order of magnitude Θ(max{D(n,p),p(1−p)}nk−2) provided that p(1−p)n1∕2→∞.
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