If $H=(V,E)$ is a 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?

anychromatically rigid graph. Do you have an example where $E_0$doesexist? $\endgroup$ – Wojowu Sep 17 '19 at 10:02infinitechromatically rigid graph. $\endgroup$ – Wojowu Sep 17 '19 at 10:12