Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). The *chromatic number* $\chi(H)$ is the smallest cardinal $\kappa \leq |V|$ such that there is a map $c:V\to \kappa$ such that for all $e\in E$ having more than one element, the restriction $c|\_e: e\to \kappa$ is not constant.


**Question.** If $H=(V,E)$ is a hypergraph with $V\neq \emptyset$ and $\kappa < \chi(H)$ is a non-empty cardinal, is there necessarily $E\_0\subseteq E$ such that $\chi(V, E\_0) = \kappa$?



Is the chromatic number of hypergraphs downward continuous?
Fred Galvin had conjectured that the answer is "yes" for *graphs* in [1] (conjecture 2), in his paper he showed that the variation of the problem to *induced* graphs is consistently false: Assume $2^{\aleph\_0}=2^{\aleph\_1}<2^{\aleph\_2}$ there exists a graph $(V,E)$ with a chromatic number of $\aleph\_2$ but for no subset $V'\subseteq V$ we have that the chromatic number of $(V', E\cap [V']^2)$ is $\aleph\_1$.


Later in [2] Péter Komjáth had shown the consistency of the failure of the above conjecture: he gave a model in which $2^{\aleph\_0}=2^{\aleph\_2}=\aleph\_3$ where there exists a graph with chromatic number of $\aleph\_2$ but no subgraph with chromatic number $\aleph\_1$.


Lastly, in [3] Shelah showed a partial positive direction: assume $V=L$, then for any $(V,E)$ and $\kappa$ such that $|V|<\kappa^{+\kappa}$ and $\chi(V,E)>\kappa$, then there exists a subgraph of chromatic number exactly $\kappa$. Further more, he had showed that under $V=L$ there is no counterexample of cardinality $\aleph\_2$: every graph with cardinality $\aleph\_2$ contains a subgraph in any smaller chromatic number.



> 
> [1] *Galvin, F.*, [**Chromatic numbers of subgraphs**](https://doi.org/10.1007/BF02276099), Period. Math. Hung. 4, 117-119 (1973). [ZBL0278.05105](https://zbmath.org/?q=an:0278.05105).
> 
> 
> 



> 
> [2] *Komjáth, Péter*, [**Consistency results on infinite graphs**](https://doi.org/10.1007/BF02772573), Israel J. Math. 61, No. 3, 285-294 (1988). [ZBL0668.05031](https://zbmath.org/?q=an:0668.05031).
> 
> 
> 



> 
> [3] *Shelah, Saharon*, Incompactness for chromatic numbers of graphs, A tribute to Paul Erdős, 361-371 (1990). [ZBL0727.05025](https://zbmath.org/?q=an:0727.05025).
> 
> 
> 


