On supersaturation in the Erdős--Sós problem

Abstract

The following classical question in extremal set theory is due to Erd\H os and Sós: what is the size of the largest family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ with no two sets $F_1,F_2\in \mathcal{F}$ such that $|F_1\cap F_2|=t$? In this paper, we address a supersaturation question for this extremal function. For a family $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ of a fixed size $\ell$, what is the smallest number of pairs $F_1,F_2\in \mathcal{F}$ with $|F_1\cap F_2|=t$ it may induce? For fixed $k$ and $n\to \infty$, we find the exact threshold when the minimum number of pairs matches the expected number of pairs in a random $\ell$-element family up to a constant factor. We also find an exact answer for $\ell$ slightly above the extremal function.