Extremal $t$-intersecting Families of Permutations for Large $t$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910255619768320 |
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| author | Saengrungkongka, Pitchayut |
| author_facet | Saengrungkongka, Pitchayut |
| contents | A set of permutations of $\{1,2,\dots,n\}$ is $t$-intersecting if any two permutations agree on at least $t$ inputs. A recent work by Kupavskii, in the spirit of the Erdős-Ko-Rado Theorem, shows that for all $t\leq n-O\left(\frac{n\log\log n}{\log n}\right)$, every $t$-intersecting family of permutations of $\{1,2,\dots,n\}$ with the maximum size must be isomorphic to the set $$A_k = \{σ: σ(i)=i\text{ for at least } t+k \text{ indices } i\in\{1,2,\dots,t+2k\}\}$$ for some $k$. By refining Kupavskii's spread approximation technique, we prove that this conclusion holds for a wider range of $t\leq n-n^{5/7+\varepsilon}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26051 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extremal $t$-intersecting Families of Permutations for Large $t$ Saengrungkongka, Pitchayut Combinatorics 05D05 A set of permutations of $\{1,2,\dots,n\}$ is $t$-intersecting if any two permutations agree on at least $t$ inputs. A recent work by Kupavskii, in the spirit of the Erdős-Ko-Rado Theorem, shows that for all $t\leq n-O\left(\frac{n\log\log n}{\log n}\right)$, every $t$-intersecting family of permutations of $\{1,2,\dots,n\}$ with the maximum size must be isomorphic to the set $$A_k = \{σ: σ(i)=i\text{ for at least } t+k \text{ indices } i\in\{1,2,\dots,t+2k\}\}$$ for some $k$. By refining Kupavskii's spread approximation technique, we prove that this conclusion holds for a wider range of $t\leq n-n^{5/7+\varepsilon}$. |
| title | Extremal $t$-intersecting Families of Permutations for Large $t$ |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2605.26051 |