Three-Layer Intersecting Temperate Families

Abstract

In response to a conjecture of Petr and Turek, we prove that every largest intersecting temperate collection of sets $\mathcal{A}\subseteq [2k{]}^{(k)}\cup [2k{]}^{(k+1)}\cup [2k{]}^{(k+2)}$ includes all $k$-sets containing some given $a\in [2k]$, all $(k+1)$-sets, and all $(k+2)$-sets not containing $a$.