Ramsey games near the critical threshold

A well-known result of Rodl and Rucinski states that for any graphHthere exists a constantCsuch that ifp >= Cn-1/m2(H), then the random graphG(n, p)is a.a.s. H-Ramsey, that is, any 2-coloring of its edges contains a monochromatic copy ofH. Aside from a few simple exceptions, the corresponding 0-statement also holds, that is, there existsc > 0 such that wheneverp <= cn-1/m2(H)the random graphG(n, p)is a.a.s. notH-Ramsey. We show that near this threshold, even whenG(n, p)is notH-Ramsey, it is often extremely close to beingH-Ramsey. More precisely, we prove that for any constantc > 0 and any strictly 2-balanced graphH, ifp >= cn-1/m2(H), then the random graphG(n, p)a.a.s. has the property that every 2-edge-coloring without monochromatic copies ofHcannot be extended to anH-free coloring after omega(1)extra random edges are added. This generalizes 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-color case and show that these theorems need not hold whenHis not strictly 2-balanced.