CDAM: Computational, Discrete and Applicable Mathematics@LSE 

CDAM Research Report, LSECDAM200816September 2008 
The 3colored Ramsey Number of Odd Cycles
Yoshiharu Kohayakawa, Miklós Simonovits, and Jozef Skokan
Denote by R(L, L, L) the minimum integer N such that any 3coloring of the edges of the complete graph K_{N} contains a monochromatic copy of a graph L. Bondy and Erdös conjectured that for an odd cycle on n vertices C_{n},Luczak proved that if n is odd, then R(C_{n}, C_{n}, C_{n}) = 4n+o(n), as n > ∞. We prove here the exact BondyErdös conjecture for sufficiently large values of n. We also describe the Ramseyextremal colorings and prove some related stability theorems.
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