On Some Exact Values of Three-Color Ramsey Numbers for Paths

For graphs G1, G2, G3, the three-color Ramsey number R(G1, G2, G3) is the smallest integer 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 Gi in color i, for some 1 ≤ i ≤ 3. First, we prove that the conjectured equality R3(C2n, C2n, C2n) = 4n, if true, implies that R3(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+(n mod 2) which was proved for sufficiently large n by Gyárfás, Ruszinkó, Sárközy, and Szemerédi in 2007. This provides more evidence for the conjecture that the latter holds for all n ≥ 1.