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 p ≥ C n − 1 / m 2 ( H ) , 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 p ≤ c n − 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‐coloring without monochromatic copies of H cannot be extended to an H ‐free coloring after ω ( 1 ) 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.