On Some Three-Color Ramsey Numbers for Paths

For graphs $G_1,G_2,G_3$, the three-color Ramsey number $R(G_1,$ $G_2, G_3)$isthesmallestinteger$n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with 3 colors, then it contains a monochromatic copy of $G_i$ in color $i$, for some $1\le i\le 3$. First, we prove that the conjectured equality $R_3(C_{2n},C_{2n},C_{2n})=4n$, if true, implies that $R_3(P_{2n+1},P_{2n+1},P_{2n+1})=4n+1$ for all $n\ge 3$. We also obtain two new exact values $R(P_8,P_8,P_8)=14$ and $R(P_9,P_9,P_9)=17$, furthermore we do so without help of computer algorithms. Our results agree with a formula $R(P_n,P_n,P_n)=2n-2+(n2)$ which was proved for sufficiently large $n$ by Gy'arf'as, Ruszink'o, S'ark"ozy, and Szemer'{e}di in 2007. This provides more evidence for the conjecture that the latter holds for all $n\ge 1$.