CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2008-16

September 2008

The 3-colored 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 3-coloring of the edges of the complete graph KN contains a monochromatic copy of a graph L. Bondy and Erdös conjectured that for an odd cycle on n vertices Cn,
R(Cn, Cn, Cn) = 4n-3 for n>3.
This is sharp if true.

Luczak proved that if n is odd, then R(Cn, Cn, Cn) = 4n+o(n), as n -> ∞. We prove here the exact Bondy-Erdös conjecture for sufficiently large values of n. We also describe the Ramsey-extremal colorings and prove some related stability theorems.

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