CDAM: Computational, Discrete and Applicable Mathematics@LSE
CDAM Research Report, LSE-CDAM-2008-17
September 2008
CDAM: Computational, Discrete and Applicable Mathematics@LSE |
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CDAM Research Report, LSE-CDAM-2008-17September 2008 |
The 3-colored Ramsey Number of Even Cycles
Fabricio Siqueira Benevides 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 on N vertices contains a monochromatic copy of a graph L. Bondy and Erdös conjectured that when L is the cycle Cn on n vertices, R(Cn, Cn, Cn) = 4n-3 for every odd n>3. Luczak proved that if n is odd, then R(Cn, Cn, Cn)=4n+o(n), as n -> ∞, and Kohayakawa, Simonovits and Skokan confirmed the Bondy-Erdös 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(Cn, Cn, Cn)=2n+o(n), as n -> ∞. In this paper, we prove that there exists n1 such that for every even n>n1, R(Cn, Cn, Cn) = 2n.
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