CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2008-17

September 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.

A PDF file (253 kB) with the full contents of this report can be downloaded by clicking here.

Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSE-CDAM-2008-17, together with your name and postal address to:
CDAM Research Reports Series
Centre for Discrete and Applicable Mathematics
London School of Economics
Houghton Street
London WC2A 2AE, U.K.
Phone: +44(0)-20-7955 7494.
Fax: +44(0)-20-7955 6877.

Introduction to the CDAM Research Report Series.
CDAM Homepage.

Copyright © London School of Economics & Political Science 2007