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arXiv · 2602.10292

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

Andrey Kupavskii·Yakov Shubin
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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.

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Andrey Kupavskii, Yakov Shubin. 2026-02-10. On supersaturation in the Erd\H{o}s--S\'os problem. https://arxiv.org/abs/2602.10292

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