On $P_4$-intersecting families of graphs

Given a graph $F$, a family $\mathcal{F}$ of graphs on $[n]$ is \emph{$F$-intersecting} if $G\cap H$ contains a copy of $F$ for every $G,H\in \mathcal{F}$. We prove that there exists an absolute constant $\epsilon >0$ such that every $P_4$-intersecting family $\mathcal{F}$ satisfies $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le (\frac{1}{2}-\epsilon )2^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}$, which resolves a conjecture of Alon. Combined with Alon’s reduction, this proves that a graph $F$ admits $F$-intersecting families of asymptotic density $1\mathrm{/}2$ if and only if $F$ is a star forest.