Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**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)

For graphs L_{1},..., L_{k}, the Ramsey number R(L_{1}, . .
. , 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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