On some three-color Ramsey numbers

In this paper we study three-color Ramsey numbers. Let K_i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G₁, G₂ and G₃, if r(G ₁, G₂) ≥ s(G₃), then r(G₁, G ₂, 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(K _1,k, K_1,m, K_n) ≤ (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(K_k,m, K _k,m, K_n) ≤ c(n/log n)^k, and r(C _2m, C_2m, K_n) ≤ c(n/log n)^m/(m-1) for sufficiently large n.