Universality among graphs omitting a complete bipartite graph

For cardinals lambda,kappa,theta we consider the class of graphs of cardinality lambda which has no subgraph which is (kappa,theta)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak) embedding. We solve this problem completely under GCH. Under various assumptions mostly related to cardinal arithmetic we prove non-existence of universals for this problem. We also look at combinatorial properties useful for those problems concerning kappa-dense families.