Ramsey numbers for graph sets versus complete graphs

Abstract

The set Ramsey number r_n(k, (ⁿ₂) ) is the smallest integer r such that if the edges of a complete graph Kr are 2-colored, then there will be a graph with n vertices and k edges in the first color or a graph with n vertices and (ⁿ₂) (e.g. a complete graph) in the second color. For each n ≥ 3 and 1 ≤ k ≤ n, the set Ramsey numbers r_n(k, (ⁿ₂)) are determined.