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- W2964741630 abstract "A well-known result of Rodl and Ruci'nski states that for any graph $H$there exists a constant $C$ such that if $p geq C n^{- 1/m_2(H)}$, then therandom graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of itsedges contains a monochromatic copy of $H$. Aside from a few simple exceptions,the corresponding $0$-statement also holds, that is, there exists $c>0$ suchthat whenever $pleq cn^{-1/m_2(H)}$ the random graph $G_{n,p}$ is a.a.s. not$H$-Ramsey. We show that near this threshold, even when $G_{n,p}$ is not $H$-Ramsey, itis often extremely close to being $H$-Ramsey. More precisely, we prove that forany constant $c > 0$ and any strictly $2$-balanced graph $H$, if $p geq cn^{-1/m_2(H)}$, then the random graph $G_{n,p}$ a.a.s. has the property thatevery $2$-edge-colouring without monochromatic copies of $H$ cannot be extendedto an $H$-free colouring after $omega(1)$ extra random edges are added. Thisgeneralises a result by Friedgut, Kohayakawa, Rodl, Ruci'nski and Tetali,who in 2002 proved the same statement for triangles, and addresses a questionraised by those authors. We also extend a result of theirs on the three-colourcase and show that these theorems need not hold when $H$ is not strictly$2$-balanced." @default.
- W2964741630 created "2019-08-13" @default.
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- W2964741630 date "2019-08-08" @default.
- W2964741630 modified "2023-09-23" @default.
- W2964741630 title "Ramsey games near the critical threshold" @default.
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