Some three-color Ramsey numbers, R(P4,P5,Ck)R(P4,P5,Ck) and R(P4,P6,Ck)R(P4,P6,Ck)

For given graphs G1,G2,G3G1,G2,G3, the three-color Ramsey number R(G1,G2,G3)R(G1,G2,G3) is defined to be the least positive integer nn such that every 3-coloring of the edges of complete graph KnKn contains a monochromatic copy of GiGi colored with ii, for some 1≤i≤31≤i≤3. In this paper, we prove that R(P4,P5,C3)=11R(P4,P5,C3)=11, R(P4,P5,C4)=7R(P4,P5,C4)=7, R(P4,P5,C5)=11R(P4,P5,C5)=11, R(P4,P5,C7)=11R(P4,P5,C7)=11, R(P4,P5,Ck)=k+2R(P4,P5,Ck)=k+2 for k≥23k≥23; R(P4,P6,C4)=8R(P4,P6,C4)=8, R(P4,P6,C3)=R(P4,P6,C5)=R(P4,P6,C7)=13R(P4,P6,C3)=R(P4,P6,C5)=R(P4,P6,C7)=13, R(P4,P6,Ck)=k+3R(P4,P6,Ck)=k+3 for k≥18k≥18.