Generalized Split Graphs and Ramsey Numbers

A graphGis called a (p, q)-split graph if its vertex set can be partitioned intoA, Bso that the order of the largest independent set inAis at mostpand the order of the largest complete subgraph inBis at mostq. Applying a well-known theorem of Erdős and Rado forΔ-systems, it is shown that for fixedp,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 numbersR(s, t) through the inequalities2F(F(R(p+2, q+2)))+1⩾f(p, q)⩾R(p+2, q+2)−1whereF(t) is the smallest number oft-element sets ensuring at+1-elementΔ-system. Apart fromf(1, 1)=5, all values off(p, q) are unknown.