Universal graphs without large bipartite graphs assuming GCH

Among a family of graphs H a graph G is called universal if any graph in H is isomorphic to an induced subgraph of G, and is called w-universal if any graph in H is isomorphic to a subgraph of G. The problem of the existence of universal and w-universal graphs was examined first for the family {G: G is of cardinality λ and omits the complete graph on κ} and was completely settled by Komjáth and Shelah (J. Combin. Theory Ser. B 63(1) (1995)). assuming GCH. The study of the families Hλ,κ,θ={G: G is a graph of cardinality λ and omits the bipartite graph B(κ,θ)} started by Komjáth and Pach (Mathematika 31 (1984)) where it was proved that there is no w-universal graph for Hℵ1,ℵ1,ℵ0 assuming ♢w1. In an unpublished result, Saharon Shelah weakened this condition to CH and his proof is presented here as Theorem 1. In Theorem 2 we replace this assumption by ♣(ℵ1) and a=ℵ1 and in Theorems 3 and 4 we give sufficient conditions for the nonexistence of a universal graph for such families (for example for H2ℵ0,2ℵ0,ℵ0 this is true in ZFC).