Ramsey games near the critical threshold

A well-known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if (Formula presented.), then the random graph G_{n, p} is a.a.s. H-Ramsey, that is, any 2-coloring 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 (Formula presented.) 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 (Formula presented.), then the random graph G_{n, p} a.a.s. has the property that every 2-edge-coloring without monochromatic copies of H cannot be extended to an H-free coloring after (Formula presented.) extra random edges are added. This generalizes a result by Friedgut, Kohayakawa, Rödl, Ruciński, 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-color case and show that these theorems need not hold when H is not strictly 2-balanced.