2-intersecting permutations

Abstract: In this paper we consider the Erd\H{o}s-Ko-Rado property for both $2$-pointwise and $2$-setwise intersecting permutations. Two permutations $\sigma,\tau \in Sym(n)$ are $t$-setwise intersecting if there exists a $t$-subset $S$ of ${1,2,\dots,n}$ such that $S^\sigma = S^\tau$. If for each $s\in S$, $s^\sigma = s^\tau$, then we say $\sigma$ and $\tau$ are $t$-pointwise intersecting. We say that $Sym(n)$ has the $t$-setwise (resp. $t$-pointwise) intersecting property if for any family $\mathcal{F}$ of $t$-setwise (resp. $t$-pointwise) intersecting permutations, $|\mathcal{F}| \leq (n-t)!t!$ (resp. $|\mathcal{F}| \leq (n-t)!$). Ellis (["Setwise intersecting families of permutation". { Journal of Combinatorial Theory, Series A}, 119(4):825-849, 2012.]), proved that for $n$ sufficiently large relative to $t$, $Sym(n)$ has the $t$-setwise intersecting property. Ellis also conjuctured that this result holds for all $n \geq t$. Ellis, Friedgut and Pilpel [Ellis, David, Ehud Friedgut, and Haran Pilpel. "Intersecting families of permutations." {Journal of the American Mathematical Society} 24(3):649-682, 2011.] also proved that for $n$ sufficiently large relative to $t$, $Sym(n)$ has the $t$-pointwise intersecting property. It is also conjectured that $Sym(n)$ has the $t$-pointwise intersecting propoperty for $n\geq 2t+1$. In this work, we prove these two conjectures for $Sym(n)$ when $t=2$.