Ramsey games near the critical threshold. Issue 4 (17th October 2020) - Conlon, David; Das, Shagnik; Lee, Joonkyung; Mészáros, Tamás
Abstract : 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.