TITLE: Universal graphs on higher cardinals
QUESTION [5 upvotes]: The Rado graph contains every finite graph as induced subgraph, and its also holds for countable graphs. So it is an universal graph of size $\aleph_0$, which contains all graphs of size $\aleph_0$ as induced subgraph. 
Question: Is it also true for higher cardinals? Does there exist a graph of size $\kappa$, which contains all graphs of size $\kappa$ as induced subgraph? If not, what is the condition for $\kappa$?

REPLY [8 votes]: An old result of Shelah states that if one adds $\lambda > \kappa^{++}$ Cohen subsets to a cardinal satisfying $\kappa^{<\kappa}$, then in the resulting universe the universality number for graphs on $\kappa^+$ is $\lambda$. It is possible to have a universal graph of size $\aleph_1$ even when CH fails (Mekler and Shelah independently).