CDAM: Computational, Discrete and Applicable Mathematics@LSE 

CDAM Research Report, LSECDAM200817September 2008 
The 3colored Ramsey Number of Even Cycles
Fabricio Siqueira Benevides and Jozef Skokan
Denote by R(L, L, L) the minimum integer N such that any 3coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdös conjectured that when L is the cycle C_{n} on n vertices, R(C_{n}, C_{n}, C_{n}) = 4n3 for every odd n>3. Luczak proved that if n is odd, then R(C_{n}, C_{n}, C_{n})=4n+o(n), as n > ∞, and Kohayakawa, Simonovits and Skokan confirmed the BondyErdös conjecture for all sufficiently large values of n.Figaj and Luczak determined an asymptotic result for the `complementary' case where the cycles are even: they showed that for even n, we have R(C_{n}, C_{n}, C_{n})=2n+o(n), as n > ∞. In this paper, we prove that there exists n_{1} such that for every even n>n_{1}, R(C_{n}, C_{n}, C_{n}) = 2n.
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