Given an infinite cardinal $\kappa$, is there a graph on $\kappa$ vertices that contains $2^\kappa$ pairwise non-isomorphic [induced subgraphs](https://en.wikipedia.org/wiki/Induced_subgraph)?



Infinite graph with lots of non-isomorphic induced subgraphs
You don't need something as complicated as the Rado graph.


Take the disjoint union of $K\_n$ (the complete graph on $n$ vertices) for all $n$. This has countably many vertices. For any subset of the natural numbers $S \subseteq \mathbb{N}$ consider the induced subgraph consisting of $K\_i$ for $i \in S$. For distinct S these subgraphs are non-isomorphic. There are uncountably many such subsets, so you are done. This is not a very interesting example of course. Perhaps you want connected induced subgraphs? But then you can just take the complement of this example.


This only answers the question for $\kappa = \aleph\_0$ of course.


