$3$-setwise intersecting families of the symmetric group

Abstract

Given two positive integers $n\ge 3$ and $t\le n$, the permutations $\sigma ,\pi \in \mathrm{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\dots ,n\}$. A family $\mathcal{F}\subset \mathrm{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\le n$ and $\mathcal{F}\subset \mathrm{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\le t!(n-t)!$ and equality holds only if $\mathcal{F}$ is a coset of a setwise stablizer of a $t$-subset of $\{1,2,\dots ,n\}$. In this paper, we prove that if $n\ge 11$ and $\mathcal{F}$ is $3$-setwise intersecting, then $|\mathcal{F}|\le 6(n-3)!$. Moreover, we prove that the characteristic vector of a $3$-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of $\mathrm{Sym}(n)$ acting on the $3$-subsets of $\{1,2,\dots ,n\}$.

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