Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 03, 2005, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Jozef Skokan (Univ. of Illinois at Urbana-Champaign and Univ. de São Paulo)

On a theorem of Luczak

For graphs L₁,..., L_k, the Ramsey number R(L₁, . . . , L_k) is the minimum integer N such that, in any coloring of the edges of the complete graph on N vertices by k colors, there exists a color i for which the corresponding color class contains L_i as a subgraph. Let C_n denote the cycle of length n. In 1973, Bondy and Erdős conjectured that if n is odd, then R(C_n,C_n,C_n) =4n-3. A great deal later, in 1999, a breakthrough was finally achieved by Luczak: he proved that R(C_n,C_n,C_n) = (4+o(1))n, where o(1) tends to 0 as n tends to infinity. In this talk, we shall outline a proof of the conjecture for all large enough n. This proof is heavily inspired on Luczak's proof, but gives the sharper result by exploiting certain stability results for extremal colourings.

Some recent related developments due to other authors will be discussed.

Joint work with M. Simonovits (Budapest) and Y. Kohayakawa (São Paulo).

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