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Source. Ruiliang Li, On an Erdős--Lovász problem: 3-critical 3-graphs of minimum degree 7, arXiv:2512.24850v1 (31 December 2025), Lemma 4.4 and proof, printed pp. 9--10 (PDF pp. 9--10).

Setup. The construction (5) of Theorem 4.1.

Dependencies. Lemma 4.3.

Used in. Theorem 1.2.

Bears on. #834: a step in the example behind the yes answer under the chromatic reading of "$3$-critical".

Statement

The hypergraph $H$ of Theorem 4.1 admits a proper 3-coloring. Hence $\chi(H)=3$.

Rewritten proof

Use the three color classes

$$\{1,2,4,5\},\qquad \{3,6,8,9\},\qquad \{7\}. \tag{1}$$

Direct comparison with the 22-edge list shows that none of its triples is contained in a class in (1), so this is a proper weak 3-coloring and $\chi(H)\leq3$. Lemma 4.3 gives $\chi(H)>2$. Hence $\chi(H)=3$.