On some three-color Ramsey numbers for paths

First, we prove that the conjectured equality R(C2n,C2n,C2n)=4n, if true, implies that R(P2n+1,P2n+1,P2n+1)=4n+1 for all n≥3. We also obtain two new exact values R(P8,P8,P8)=14 and R(P9,P9,P9)=17, furthermore we do so without help of computer algorithms. Our results agree with a formula R(Pn,Pn,Pn)=2n−2+(nmod2) which was proved for sufficiently large n by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007). This provides more evidence for the conjecture that the latter holds for all n≥1.