Abstract
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 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 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 , 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 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.
Highlights
The study of sparse generalizations of combinatorial theorems has attracted considerable interest in recent years and there are several general mechanisms [2, 3, 16, 17] that allow one to prove that analogues of classical results such as Ramsey’s theorem, Turán’s theorem and Szemerédi’s theorem hold relative to sparse random graphs and sets of integers
Much of this work is based, in one way or another, on the beautiful random Ramsey theorem of Rödl and Rucinski [14, 15] from 1995. This seminal result gives a complete answer to the question of when the binomial random graph Gn,p is (H, r)-Ramsey, that is, has the property that any r-coloring of its edges contains a monochromatic copy of the graph H
If q = O(n−1∕m(H)), with positive probability Gn,q is H-free, giving a valid extension of φ3. Note that these results cannot be extended to r ≥ 4 colors, since the two random graphs Gn,p and Gn,q can be colored independently with disjoint pairs of colors, so we can avoid creating a monochromatic copy of H until the density of one of the two random graphs exceeds the random Ramsey threshold Cn−1∕m2(H) from Theorem 1
Summary
The study of sparse generalizations of combinatorial theorems has attracted considerable interest in recent years and there are several general mechanisms [2, 3, 16, 17] that allow one to prove that analogues of classical results such as Ramsey’s theorem, Turán’s theorem and Szemerédi’s theorem hold relative to sparse random graphs and sets of integers. Part (a) of the theorem above says that for any c > 0, no matter how small, if p = cn−1∕2, even though there are 2-edge-colorings of Gn,p containing no monochromatic K3, no such coloring can be extended to a monochromatic-K3-free 2-edge-coloring after ω(1) extra random edges are added One interpretation of this result is that for any c > 0 the random graph Gn,p with p = cn−1∕2 is, with high probability, already extremely close to being (K3, 2)-Ramsey. If q = O(n−1∕m(H)), with positive probability Gn,q is H-free, giving a valid extension of φ3 Note that these results cannot be extended to r ≥ 4 colors, since the two random graphs Gn,p and Gn,q can be colored independently with disjoint pairs of colors, so we can avoid creating a monochromatic copy of H until the density of one of the two random graphs exceeds the random Ramsey threshold Cn−1∕m2(H) from Theorem 1
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