On some Ramsey and Turan-type numbers for paths and cycles

For given graphs G(1),G(2),...,G(k), where k >= 2, the multicolor Ramsey number R(G(1),G(2),...,G(k)) 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 G(i) colored with i, for some 1 <= i <= k. Let P-k (resp. C-k) be the path (resp. cycle) on k vertices. In the paper we show that R(P-3,C-k,C-k)=R(C-k,C-k)=2k-1 for odd k. In addition, we provide the exact values for Ramsey numbers R(P-4,P-4,C-k)=k+2 and R(P-3,P-5,C-k)=k+1.