On generalized Ramsey numbers

Let f1 and f2 be graph parameters. The Ramsey number r(f1⩾m;f2⩾n) is defined as the minimum integer N such that any graph G on N vertices, either f1(G)⩾m or f2(Ḡ)⩾n. A general existence condition is given and a general upper bound is shown in this paper. In addition, suppose the number of triangles in G is denoted by t(G). We verify that (1−o(1))(24n)1/3⩽r(t⩾n;t⩾n)⩽(1+o(1))(48n)1/3 as n→∞.