A well‐known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if urn:x-wiley:rsa:media:rsa20959:rsa20959-math-0001, then the random graph Gn, 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 urn:x-wiley:rsa:media:rsa20959:rsa20959-math-0002 the random graph Gn, p is a.a.s. not H‐Ramsey. We show that near this threshold, even when Gn, 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 urn:x-wiley:rsa:media:rsa20959:rsa20959-math-0003, then the random graph Gn, 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 urn:x-wiley:rsa:media:rsa20959:rsa20959-math-0004 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.