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- W3138130638 abstract "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." @default.
- W3138130638 created "2021-03-29" @default.
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- W3138130638 date "2021-03-23" @default.
- W3138130638 modified "2023-10-18" @default.
- W3138130638 title "$K_4$-intersecting families of graphs" @default.
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- W3138130638 doi "https://doi.org/10.48550/arxiv.2103.12671" @default.
- W3138130638 hasPublicationYear "2021" @default.
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