On some three color Ramsey numbers for paths and cycles

For given graphs G1,G2,…,Gk, k≥2, the multicolor Ramsey number, denoted by R(G1,G2,…,Gk), is the smallest integer n such that if we arbitrarily color the edges of a 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 consider the value for numbers of type R(Pi,Pk,Cm) for odd m, k≥m≥3 and k>3i2−14i+254 when i is odd, and k>3i2−10i+208 when i is even. In addition, we provide the exact values for Ramsey numbers R(P3,Pk,C4) for all integers k≥3.