  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 nonexistence
of universals for this problem and some related ones.
