Perhaps the best known concept in Extremal Graph Theory is the Ramsey number. A red-blue coloring of a graph Gassigns every edge of G the color red or blue. For two graphs F and H, the Ramsey number R(F, H) of F and H is the smallest positive integer n such that for every red-blue coloring of the complete graph K n of order n, there is either a subgraph F all of whose edges are colored red (are d F) or a subgraph H all of whose edges are colored blue (a blue H). In the case where F and H are bipartite (those having no cycles with an odd number of vertices), the bipartite Ramsey number BR(F, H) was defined by Beineke and Schwenk in 1975 as the smallest positive integer r such that every red-blue coloring of the r-regular complete bipartite graph Kr, r results in a red For a blue H.

We look at an extension of this idea. For an integer k ≥ 2, a balanced complete k-partite graph of order n≥k is the complete k-partite graph in which every partite set has either bn/kc or dn/ke vertices. For bipartite graphs F and H and an integer k with 2≤k≤R(F, H), the k-Ramsey number Rk(F, H) was defined by Chart r and and Zhang in 2013 as the smallest positive integer n such that every red-blue coloring of a balanced complete k-partite graph of order n produces either a red F or a blue H. This concept provides a new look at the well-known concept of Ramsey numbers and gives rise to some interesting questions.