On supersaturation in the Erdős--Sós problem
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908827462402048 |
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| author | Kupavskii, Andrey Shubin, Yakov |
| author_facet | Kupavskii, Andrey Shubin, Yakov |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10292 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On supersaturation in the Erdős--Sós problem Kupavskii, Andrey Shubin, Yakov Combinatorics 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. |
| title | On supersaturation in the Erdős--Sós problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.10292 |