### Universality among graphs omitting a complete bipartite graph

by Shelah. [Sh:706]

Combinatorica, 2012

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.

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