On $r$-wise $t$-intersecting uniform families
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910624233029632 |
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| author | Frankl, Peter Wang, Jian |
| author_facet | Frankl, Peter Wang, Jian |
| contents | We consider families, $\mathcal{F}$ of $k$-subsets of an $n$-set. For integers $r\geq 2$, $t\geq 1$, $\mathcal{F}$ is called $r$-wise $t$-intersecting if any $r$ of its members have at least $t$ elements in common. The most natural construction of such a family is the full $t$-star, consisting of all $k$-sets containing a fixed $t$-set. In the case $r=2$ the Exact Erdős-Ko-Rado Theorem shows that the full $t$-star is largest if $n\geq (t+1)(k-t+1)$. In the present paper, we prove that for $n\geq (2.5t)^{1/(r-1)}(k-t)+k$, the full $t$-star is largest in case of $r\geq 3$. Examples show that the exponent $\frac{1}{r-1}$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $r$-wise $t$-intersecting uniform families Frankl, Peter Wang, Jian Combinatorics We consider families, $\mathcal{F}$ of $k$-subsets of an $n$-set. For integers $r\geq 2$, $t\geq 1$, $\mathcal{F}$ is called $r$-wise $t$-intersecting if any $r$ of its members have at least $t$ elements in common. The most natural construction of such a family is the full $t$-star, consisting of all $k$-sets containing a fixed $t$-set. In the case $r=2$ the Exact Erdős-Ko-Rado Theorem shows that the full $t$-star is largest if $n\geq (t+1)(k-t+1)$. In the present paper, we prove that for $n\geq (2.5t)^{1/(r-1)}(k-t)+k$, the full $t$-star is largest in case of $r\geq 3$. Examples show that the exponent $\frac{1}{r-1}$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz. |
| title | On $r$-wise $t$-intersecting uniform families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.19344 |