On supersaturation in the Erd\H{o}s--S\'os problem

TL;DR

This paper investigates the supersaturation phenomenon in the Erdős–Sós problem, determining the minimal number of intersecting pairs in large families of sets and identifying thresholds where these numbers match random expectations.

Contribution

It provides exact thresholds and counts for the minimal number of intersecting pairs in families of sets, extending classical extremal set theory results.

Findings

Exact threshold for the minimal number of intersecting pairs

Matching the expected number of pairs in random families

Results for families slightly larger than the extremal size

Abstract

The following classical question in extremal set theory is due to Erd\H os and S\'os: what is the size of the largest family $\mathcal F\subset {[n]\choose k}$ 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 $\mathcal F\subset {[n]\choose k}$ 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.