A Problem on Complements and Disjoint Edges in a Hypergraph

Let KN denote the complete k-graph on N vertices, that is, the k-uniform hypergraph whose edges are the (k) k-subsets of a set of N vertices . The Ramsey number of a pair of k-graphs is defined as for graphs in terms of the 2-colorings of the k-sets which are the edges of a complete k-graph . Let sKk denote a k-graph which is a matching with s edges . That is, sKk has sk vertices and s edges, no two of which intersect . The question of determining the Ramsey number of a pair of k-graphs, one of which is a matching, arose in connection with work done by the first author and S . Burr [1] . (The Ramsey number of a pair of graphs, one of which being a matching, recently been Equation (1)