On supersaturation in the Erdős--Sós problem

The following classical question in extremal set theory is due to Erd\H os and Sós: what is the size of the largest family $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ with no two sets $F_1,F_2\in \mathcal{F}$ such that $\mathrm{\mid }{F}_1\cap F_2\mathrm{\mid }=t$? In this paper, we address a supersaturation question for this extremal function. For a family $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ of a fixed size $\mathrm{\ell }$, what is the smallest number of pairs $F_1,F_2\in \mathcal{F}$ with $\mathrm{\mid }{F}_1\cap F_2\mathrm{\mid }=t$ it may induce? For fixed $k$ and $n\to \mathrm{\infty }$, we find the exact threshold when the minimum number of pairs matches the expected number of pairs in a random $\mathrm{\ell }$-element family up to a constant factor. We also find an exact answer for $\mathrm{\ell }$ slightly above the extremal function.