For a family of graphs $mathcal{H}$ and an integer k, we denote by $R^k(mathcal{H})$ the corresponding k-Ramsey number, which is defined to be the smallest integer n such that every k-coloring of the edges of $K_n$ contains a monochromatic copy of a graph in $mathcal{H}$. The local k-Ramsey number $R^k_{loc}(mathcal{H})$ and the mean k-Ramsey number $R^k_{mean}(mathcal{H})$ are defined analogously. Let $mathcal{G}$ be the family of non-bipartite graphs and $T_n$ be the family of all trees on n vertices. In this paper we prove that $R^k_{loc}(mathcal{G})=R^k_{mean}(mathcal{G})$, and $R^2(T_n)$ < $R^2_{loc}(T_n)4 = $R^2_{mean}(T_n)$ for all $n;{ge};3$.