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arXiv · 2605.26051

Extremal $t$-intersecting Families of Permutations for Large $t$

Pitchayut Saengrungkongka
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Abstract

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 Erd\H{o}s-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 $$A_k = \{\sigma : \sigma(i)=i\text{ for at least } t+k \text{ indices } i\in\{1,2,\dots,t+2k\}\}$$ for some $k$. By refining Kupavskii's spread approximation technique, we prove that this conclusion holds for a wider range of $t\leq n-n^{5/7+\varepsilon}$.

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Pitchayut Saengrungkongka. 2026-05-25. Extremal $t$-intersecting Families of Permutations for Large $t$. https://arxiv.org/abs/2605.26051

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