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arXiv · 2610.32677

Sharp asymmetric vertex-Ramsey properties of graphs

Robert Hancock·Joseph Hyde·Mael Kupperschmitt
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Abstract

In 1992, Łuczak, Ruciński and Voigt showed that for any graph $H$, a parameter $m_1(H)$ - called the 1-density - governs the threshold at which the binomial random graph $G(n,p)$ inherits the vertex-Ramsey property. In 2000, Friedgut and Krivelevich showed this threshold is sharp for graphs satisfying a natural balancedness condition. In 1996, Kreuter showed that for any graphs $H_1, \ldots, H_r$ with $m_1(H_1) \geq \cdots \geq m_1(H_r)$, a parameter dependent only on $H_1$ and $H_2$ governs the threshold at which $G(n,p)$ inherits the `asymmetric' vertex-Ramsey property. We prove this threshold is sharp for graphs respecting natural balancedness conditions, including in the analogous hypergraph setting.

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Robert Hancock, Joseph Hyde, Mael Kupperschmitt. 2026-10-23. Sharp asymmetric vertex-Ramsey properties of graphs. https://arxiv.org/abs/2610.32677

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