4 problems

Let $obreakboldsymbol{text{?}}$...

Let $S_n$ be the symmetric group. A family $\mathcal{A}\subset S_n$ is reversing if for every $\sigma,\pi\in\mathcal{A}$ there is a pair $\\{i,j\\}\subseteq[n]$ such that…

Double-translate conjecture. A maximum-sized $t$-intersecting family in $S_n$ must be of the form $\pi M_i\tau$ for some $i$ in this range and some $\pi,\tau\in S_n$. The source st…

Let $S_n$ be the symmetric group. A family $\mathcal{A}\subset S_n$ is $t$-set-intersecting if for every $\sigma,\pi\in\mathcal{A}$ there is a $t$-element subset $x\subseteq[n]$ su…