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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.05704 |
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| _version_ | 1866913881121619968 |
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| author | Frankl, Peter Wang, Jian |
| author_facet | Frankl, Peter Wang, Jian |
| contents | Let $n>2r>0$ be integers. We consider families $\mathcal{F}$ of subsets of an $n$-element set, in which the union of any two members has size at most $2r$. One of our results states that for $n\geq 6r$ the number of members of size exceeding $r$ in $\mathcal{F}$ is at most $\binom{n-2}{r-1}$. Another result shows that for $n>3.5r$ the number of sets of size at least $r$ is at most $\binom{n}{r}$. Both bounds are best possible and the latter sharpens the classical Katona Theorem. Similar results are proved for the odd case of the Katona Theorem as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The overflow in the Katona Theorem Frankl, Peter Wang, Jian Combinatorics Let $n>2r>0$ be integers. We consider families $\mathcal{F}$ of subsets of an $n$-element set, in which the union of any two members has size at most $2r$. One of our results states that for $n\geq 6r$ the number of members of size exceeding $r$ in $\mathcal{F}$ is at most $\binom{n-2}{r-1}$. Another result shows that for $n>3.5r$ the number of sets of size at least $r$ is at most $\binom{n}{r}$. Both bounds are best possible and the latter sharpens the classical Katona Theorem. Similar results are proved for the odd case of the Katona Theorem as well. |
| title | The overflow in the Katona Theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.05704 |