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Ramsey Properties of Random Graphs and Folkman Numbers

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Rödl V., Ruciński A., Schacht M.

Discussiones Mathematicae Graph Theory

2017

tom 37

nr 3

755-776

For two graphs, G and F, and an integer r ≥ 2 we write G → (F)r if every r-coloring of the edges of G results in a monochromatic copy of F. In 1995, the first two authors established a threshold edge probability for the Ramsey property G(n, p) → (F)r, where G(n, p) is a random graph obtained by including each edge of the complete graph on n vertices, independently, with probability p. The original proof was based on the regularity lemma of Szemerédi and this led to tower-type dependencies between the involved parameters. Here, for r = 2, we provide a self-contained proof of a quantitative version of the Ramsey threshold theorem with only double exponential dependencies between the constants. As a corollary we obtain a double exponential upper bound on the 2-color Folkman numbers. By a different proof technique, a similar result was obtained independently by Conlon and Gowers.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Ruciński A.

1 Rödl V.

1 Schacht M.

1 2017

Wyniki wyszukiwania

Ramsey Properties of Random Graphs and Folkman Numbers