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arXiv · 2609.13145

On $P_4$-intersecting families of graphs

Jie Han·Bin Wang
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Abstract

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 $\varepsilon>0$ such that every $P_4$-intersecting family $\mathcal F$ satisfies $|\mathcal F|\le\left(\frac12-\varepsilon\right)2^{\binom n2}$, which resolves a conjecture of Alon. Combined with Alon's reduction, this proves that a graph $F$ admits $F$-intersecting families of asymptotic density $1/2$ if and only if $F$ is a star forest.

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Jie Han, Bin Wang. 2026-09-11. On $P_4$-intersecting families of graphs. https://arxiv.org/abs/2609.13145

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