Nearly Erdős-Ko-Rado theorems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918281550495744 |
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| author | Katona, Gyula O. H. Wang, Jian |
| author_facet | Katona, Gyula O. H. Wang, Jian |
| contents | If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\leq n$ then $|\mathcal{F}|\leq {n-1\choose k-1}$ holds by the celebrated Erdős-Ko-Rado theorem. But an intersecting family obviously satisfies the condition $${\ell \choose 2}\leq \sum_{1\leq i<j\leq \ell}|F_i\cap F_j| $$ for any $\ell$ distinct members of the family. It has been proved in [5] that even if ${\ell \choose 2}$ is replaced by ${\ell -1 \choose 2}+1$ the conclusion $|\mathcal{F}|\leq {n-1\choose k-1}$ remains valid for large $n$. However the 1 cannot be omitted, because there is a larger family satisfying that weaker condition. In the present paper we determine the largest size of the family under this weaker condition when $n$ is sufficiently large. All of these are treated in the more general setting of $t$-intersecting families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06871 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nearly Erdős-Ko-Rado theorems Katona, Gyula O. H. Wang, Jian Combinatorics If a family $\mathcal{F}$ of $k$-element subsets of an $n$-element set is pairwise intersecting, $2k\leq n$ then $|\mathcal{F}|\leq {n-1\choose k-1}$ holds by the celebrated Erdős-Ko-Rado theorem. But an intersecting family obviously satisfies the condition $${\ell \choose 2}\leq \sum_{1\leq i<j\leq \ell}|F_i\cap F_j| $$ for any $\ell$ distinct members of the family. It has been proved in [5] that even if ${\ell \choose 2}$ is replaced by ${\ell -1 \choose 2}+1$ the conclusion $|\mathcal{F}|\leq {n-1\choose k-1}$ remains valid for large $n$. However the 1 cannot be omitted, because there is a larger family satisfying that weaker condition. In the present paper we determine the largest size of the family under this weaker condition when $n$ is sufficiently large. All of these are treated in the more general setting of $t$-intersecting families. |
| title | Nearly Erdős-Ko-Rado theorems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.06871 |