On supersaturation in the Erdős--Sós problem
Andrey Kupavskii, Yakov Shubin
TL;DR
This work investigates the supersaturation phenomenon for the Erdős–Sós problem on $k$-subsets of $[n]$ avoiding exact intersections of size $t$. By deploying the Delta-system (sunflower) method, together with Füredi’s structural theorems and Kruskal–Katona-type shadow lemmas, the authors determine the asymptotic order of the minimum number $\\rho(\\ell)$ of $t$-intersecting pairs in an $\\ell$-vertex family for fixed $k,t$ as $n\to\infty$, and identify exact values in several regimes. They establish an upper bound from averaging and derive sharp lower bounds, with a key threshold at the extremal size $\\alpha(G(n,k,t))$, where the random-pair expectation becomes the correct benchmark up to constants. The paper also presents two extremal constructions illustrating tightness and provides exact values of $\\rho(l)$ near the extremal size under small additive increments $r$, including a precise regime where $\\rho(l)=r\binom{k}{t}\binom{n-k-t-1}{k-2t-1}$. These results deepen understanding of supersaturation in the Erdős–Sós framework and highlight the rich structure of almost-extremal families in uniform intersection problems.
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 $\mathcal F\subset {[n]\choose k}$ 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.