Generalized Split Graphs and Ramsey Numbers

A graph G is called a ( p, q)-split graph if its vertex set can be partitioned into A, B so that the order of the largest independent set in A is at most p and the order of the largest complete subgraph in B is at most q. Applying a well-known theorem of Erdo s and Rado for 2-systems, it is shown that for fixed p, q, ( p, q)-split graphs can be characterized by excluding a finite set of forbidden subgraphs, called ( p, q)split critical graphs. The order of the largest ( p, q)-split critical graph, f ( p, q), relates to classical Ramsey numbers R(s, t) through the inequalities