Ramsey games near the critical threshold

A well-known result of Rodl and Rucinski states that for any graph H there exists a constant C such that if p ≥ Cn^(−1/m_(2)(H)), then the random graph G_(n,p) is a.a.s. H-Ramsey, that is, any 2-colouring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0-statement also holds, that is, there exists c > 0 such that whenever p ≤ 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, it is often extremely close to being H-Ramsey. More precisely, we prove that for any constant c > 0 and any strictly 2-balanced graph H, if p ≥ c^(n−1/m_(2)(H)), then the random graph G_(n,p) a.a.s. has the property that every 2-edge-colouring without monochromatic copies of H cannot be extended to an H-free colouring after ω(1) extra random edges are added. This generalises a result by Friedgut, Kohayakawa, Rodl, Rucinski and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three-colour case and show that these theorems need not hold when H is not strictly 2-balanced.