Ellis–Filmus–Friedgut theorem for clique-intersecting graph families

Let $t$ and $n$ be positive integers with $t\geq 2$, and let $\mathcal{G}$ be a family of graphs on the common labeled vertex set $[n]=\{1,\ldots,n\}$. The family is $K_t$-intersecting if the intersection of every two graphs in $\mathcal{G}$ contains a subgraph isomorphic to the complete graph $K_t$.

Ellis–Filmus–Friedgut theorem. Every $K_t$-intersecting family of graphs on $[n]$ has size at most

$$2^{\binom{n}{2}-\binom{t}{2}},$$

with equality for the family of all graphs containing a fixed clique on $t$ vertices.

This resolves the corresponding extremal problem for complete graphs and, in particular, proves the conjectured value in the triangle case. The statement is attributed in the source to Ellis, Filmus, and Friedgut.