Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.06731 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915702230745088 |
|---|---|
| author | Wang, Jian Xiao, Jimeng |
| author_facet | Wang, Jian Xiao, Jimeng |
| contents | Let $\mathcal{S}_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. A family $\mathcal{F}\subset \mathcal{S}_n$ is called intersecting if for every $σ,π\in \mathcal{F}$ there exists some $i\in [n]$ such that $σ(i)=π(i)$. Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family $\mathcal{F}$ is defined as the minimum number of permutations in $\mathcal{F}$, whose deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for $n\geq 500$ the diversity of an intersecting subfamily of $\mathcal{S}_n$ is at most $(n-3)(n-3)!$, which is best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the maximum diversity of intersecting families in the symmetric group Wang, Jian Xiao, Jimeng Combinatorics 05D05 Let $\mathcal{S}_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. A family $\mathcal{F}\subset \mathcal{S}_n$ is called intersecting if for every $σ,π\in \mathcal{F}$ there exists some $i\in [n]$ such that $σ(i)=π(i)$. Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family $\mathcal{F}$ is defined as the minimum number of permutations in $\mathcal{F}$, whose deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for $n\geq 500$ the diversity of an intersecting subfamily of $\mathcal{S}_n$ is at most $(n-3)(n-3)!$, which is best possible. |
| title | A note on the maximum diversity of intersecting families in the symmetric group |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2501.06731 |