$3$-setwise intersecting families of the symmetric group

TL;DR

This paper proves an upper bound on the size of 3-setwise intersecting families in the symmetric group for n ≥ 11, confirming a special case of a broader conjecture and characterizing maximum families via eigenstructure.

Contribution

It establishes the maximum size of 3-setwise intersecting families in Sym(n) for n ≥ 11 and characterizes their structure through eigenspaces of the permutation module.

Findings

Maximum size of 3-setwise intersecting families is 6(n-3)! for n ≥ 11.

Characteristic vectors of maximum families lie in specific eigenspaces.

Confirms a case of Ellis's conjecture for t=3.

Abstract

Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $\sigma,\pi \in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset \operatorname{Sym}(n)$ 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 $t\leq n$ and $\mathcal{F} \subset \operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\mathcal{F}|\leq t!(n-t)!$ and equality holds only if $\mathcal{F}$ is a coset of a setwise stablizer of a $t$-subset of $\{1,2,\ldots,n\}$. In this paper, we prove that if $n\geq 11$ and $\mathcal{F}$ is $3$-setwise intersecting, then $|\mathcal{F}|\leq 6(n-3)!$. 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 $\operatorname{Sym}(n)$ acting on the $3$-subsets of $\{1,2,\ldots,n\}$.