Some three-color Ramsey numbers, R(P4, P5, Ck) and R(P4, P6, Ck)

For given graphs G(1), G(2), G(3), the three-color Ramsey number R(G(1), G(2), G(3)) is defined to be the least positive integer n such that every 3-coloring of the edges of complete graph K-n contains a monochromatic copy of G(i) colored with i. for some 1 <= i <= 3. In this paper, we prove that R(P-4, P-5, C-3) = 11, R(P-4, P-5, C-4) = 7, R(P-4, P-5, C-5) = 11, R(P-4, P-5, C-7) = 11, R(P-4, P-5, C-k) = k + 2 for k >= 23; R(P-4, P-6, C-4) = 8, R(P-4, P-6, C-3) = R(P-4, P-6, C-5) = R(P-4, P-6, C-7) = 13, R(P-4, P-6, C-k) = k + 3 for k >= 18. (C) 2008 Elsevier Ltd. All rights reserved.