On Some Three-Color Ramsey Numbers

In this paper we study three-color Ramsey numbers. Let Ki,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G1, G2 and G3, if r(G1, G2) ≥ s(G3), then r(G1, G2, G3) ≥ (r(G1, G2) − 1)(χ(G3) − 1) + s(G3), where s(G3) is the chromatic surplus of G3; (ii)(k + m − 2)(n − 1) + 1 ≤ r(K1,k,K1,m,Kn) ≤ (k + m − 1)(n − 1) + 1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m ≥ k ≥ 2, there is a constant c such that r(Kk,m,Kk,m,Kn) ≤ c(n/ log n) , and r(C2m, C2m,Kn) ≤ c(n/ log n) m/(m−1) for sufficiently large n.