How unproportional must a graph be?

Abstract: Let $u_k(G,p)$ 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 $p\binom{n}{2}$ to the nearest integer. Our main result is that, for fixed $k\ge 4$ and for $n$ large, the minimum of $u_k(G,p)$ over $n$-vertex graphs has order of magnitude $\Theta\big(\max{D(n,p), p(1-p)} n^{k-2}\big)$ provided that $p(1-p)n^{1/2} \to \infty$.