Balanced subgraph counts and quasirandomness: a proof of the Janson--Sós conjecture

Abstract

Let $F$ be a finite simple graph with $m$ vertices and at least two edges, and let $p\in(0,1]$. We prove that a graph sequence is $p$-quasirandom if and only if the symmetrized counts of copies of $F$ across every balanced partition into $m$ classes have the asymptotics predicted by the random graph with edge probability $p$. This settles a conjecture of Janson and Sós for all such $F$, including graphs with isolated vertices. The proof completes their graph-limit reduction by a concavity argument for a sum of fractional powers determined by the degree sequence of $F$. We also show that a symmetrized subgraph kernel of positive mean can be additive in its coordinates only when the underlying graphon is constant almost everywhere. This rigidity result yields a quasirandomness criterion for generalized restricted subgraph counts, resolving a further conjecture of Janson.