If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kappa$ is a cardinal, then $c:V\to\kappa$ is a *coloring* if for every $e\in E$ with $|e|>1$, the restriction $c|\_e:e \to \kappa$ is non-constant. By $\chi(H)$ we denote the smallest cardinal $\kappa$ such that there is a coloring $c:V\to \kappa$. We say that a hypergraph $H=(V,E)$ is *chromatically rigid* if $\chi(H) = |V|$. 


If $H=(V,E)$ is chromatically rigid, is there always $E\_0\subseteq E$ such that $(V,E\_0)$ is chromatically rigid, and $E\_0$ is $\subseteq$-minimal with respect to this property?



Chromatically rigid hypergraphs
Let $H$ be a complete graph on $V=\omega$. Suppose $E\_0$ existed. Then removing a single edge would give a finitely colorable graph. Adding this edge back, we can color the graph with at most one more color. Same should work for any infinite $V$.


