On Some Ramsey and Turan-Type Numbers for Paths and Cycles

For given graphs G1, G2, ..., Gk, where k ≥ 2, the multicolor Ramsey number R(G1, G2, ..., Gk) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, there is always a monochromatic copy of Gi colored with i, for some 1 ≤ i ≤ k. Let Pk (resp. Ck) be the path (resp. cycle) on k vertices. In the paper we show that R(P3, Ck, Ck) = R(Ck, Ck) = 2k − 1 for odd k. In addition, we provide the exact values for Ramsey numbers R(P4, P4, Ck) = k + 2 and R(P3, P5, Ck) = k + 1.