Ramsey numbers for cycles in graphs

Given two graphs G1, G2, the Ramsey number R(G1, G2) is the smallest integer m such that, for any partition (E1, E2) of the edges of Km, either G1 is a subgraph of the graph induced by E1, or G2 is a subgraph of the graph induced by E2. We show that R(Cn, Cn)=2n-1 if n is odd, R(Cn, C2r-1)=2n-1 if n>r(2r-1), R(Cn, C2r)=n+r-1 if n>4r2-r+2, R(Cn, Kr)≤nr2 for all r,n, R(Cn, Kr)=(r-1)(n-1)+1 if n≥r2-2, R(Cn, Kt+1r)=t(n-1)+r for large n