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 $\kappa$?
Following up on Asaf's comment, indeed, if $\kappa^{\lt\kappa}=\kappa$, then we may build a $\kappa$-universal graph of size $\kappa$. Proceed in $\kappa$ many stages. At each stage, we have a graph of size $\gamma$, less than $\kappa$. There are precisely $2^\gamma\leq\kappa$ many ways to add a single new point to this graph. Using a pairing function, we add at stage $\langle\alpha,\beta\rangle$ a point realizing the $\beta^{th}$ pattern of connectivity with the points constructed at stage $\alpha$. It follows that after $\kappa$ many stages, we will have a $\kappa$-universal graph, since for any graph of size $\kappa$, we enumerate its nodes and then systematically find a copy inside our universal graph realizing that pattern of connectivity with the previous nodes. This is the "forth" part of the usual back-and-forth arguments.
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).
