Joseph Hyde

Sharp asymmetric vertex Ramsey properties of random hypergraphs

In 1992, Luczak, Rucinski and Voigt showed that for any graph $H$, a parameter $m_1(H)$ - called the 1-density - governs the threshold at which $G(n,p)$ inherits the vertex Ramsey property. In 2000, Freidgut and Krivelevich showed this threshold is sharp. 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 that this threshold is sharp (as long as the graphs respect some natural balancedness conditions), including in the analogous hypergraph setting.

Joint work with Mael Kupperschmidt, and with Robert Hancock, Matthew Jenssen and Adva Mond.