A well‐known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if , 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 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 , 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 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.

中文翻译：

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拉姆齐比赛接近临界点

Rödl和Ruciński的著名结果表明，对于任何图H，存在一个常数C，使得如果，则随机图G _{n，  p}为aas  H- Ramsey，即，其边缘的任何2种颜色都包含a H的单色副本。除了一些简单的例外，相应的0语句也成立，也就是说，存在c  > 0，因此只要随机图G _{n，  p}都不是H‐ Ramsey。我们证明即使在G _{n，  p}不是H‐ Ramsey，通常非常接近H‐ Ramsey。更确切地说，我们证明了对任何恒定ç  > 0和任何严格2平衡图表ħ，如果，则该随机图ģ _{Ñ，  p} AAS具有这样的特性：每2边着色无单色副本ħ不能延伸到无H着色后添加了额外的随机边缘。这概括了Friedgut，Kohayakawa，Rödl，Ruciński和Tetali的结果，他们在2002年证明了三角形的相同说法，并解决了这些作者提出的问题。我们还在三色情况下扩展了它们的结果，并表明当H并非严格地为2平衡时，这些定理不需要成立。