$K_4$-intersecting families of graphs

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and S\'os showing that the maximum size of a triangle-intersecting family of graphs on $n$ vertices has size at most $2^{\binom{n}{2} - 3}$, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to $K_t$-intersecting families. We prove these conjectures for $t \in \{3,4\}$, showing that if $\mathcal F_1$ and $\mathcal F_2$ are families of graphs on $n$ labeled vertices such that for any $G_1 \in \mathcal F_1$ and $G_2 \in \mathcal F_2$, $G_1 \cap G_2$ contains a $K_t$, then $\lvert \mathcal F_1 \rvert \lvert \mathcal F_2 \rvert \le 4^{\binom{n}{2} - \binom{t}{2}}$, with equality if and only if $\mathcal F_1 = \mathcal F_2$ consists of all graphs that contain some fixed $K_t$. We also establish a stability result. More generally, "$G_1 \cap G_2$ contains a $K_t$" can be replaced by "$G_1$ and $G_2$ agree on a non-$(t-1)$-colorable graph."