Let $G = (V,E)$ be a simple, undirected graph with $\chi(G)$ infinite. Given a cardinal $\kappa$ with $0 < \kappa < \chi(G)$, is there an induced subgraph $S$ of $G$ with $\chi(S) = \kappa$?


**What I tried:** Let ${\cal S}$ be the collection of all subgraphs of $G$ colorable with $\kappa$ colors. I hoped to find a maximal element $M$ in ${\cal S}$ using [Zorn's Lemma](https://en.wikipedia.org/wiki/Zorn%27s_lemma), establishing $\chi(M) = \kappa$. But this approach [does not work](https://mathoverflow.net/questions/400644/is-the-union-of-a-chain-of-kappa-colorable-subgraphs-kappa-colorable).



Induced subgraphs of any given smaller chromatic number
Not necessarily. Komjáth showed that it is consistent that there is a graph of chromatic number $\aleph\_2$ which does not have a subgraph (not just induced) of chromatic number $\aleph\_1$. See P. Komjáth, Consistency results on infinite graphs, Israel J. Math., 61 (1988), pp. 285-294.


