On Some Three-color Ramsey Numbers of Paths and Cycles

Abstract:

For given graphs G1,G2,…,Gk, where k≥2, the k-color 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 color i, for some 1≤i≤k. In this note, we provide the exact value for 3color Ramsey R(Pm , Pm,Cn), where n is larger than m.