The Janson–Sós conjecture on quasi-random properties from graph averages

Let $F \neq K_2$ be a non-empty graph with vertex set $\{1,\ldots,m\}$, and let $\alpha_1,\ldots,\alpha_m \in (0,1)$ satisfy

$$\sum_{i=1}^m \alpha_i \leqslant 1.$$

Write $\mathcal{P}(F,\alpha_1,\ldots,\alpha_m)$ for the property defined by the corresponding average over products of disjoint vertex sets, and let $p \in (0,1]$. Janson–Sós conjecture. The property $\mathcal{P}(F,\alpha_1,\ldots,\alpha_m)$ is a quasi-random property for every $p \in (0,1]$. This addresses the problem of determining when these properties are always quasi-random; before this work it was unknown even for the path on three vertices and was stated as an open problem by Janson and Sós. The exceptional known non-example is $\mathcal{P}(K_2,\frac{1}{2},\frac{1}{2})$; the claim excludes it.