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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 2.2 and proof, printed p. 4 (PDF p. 4).

Dependencies. None.

Used in. Proposition 4.5.

Bears on. #834: the certificate criterion by which the paper checks edge-criticality of its example under the chromatic reading of "$3$-critical".

Statement

Under the paper's standing convention that hypergraphs are finite, simple (with no repeated edges), and have no empty edge, consider an edge $e$ of a hypergraph $H$ with $\chi(H)\geq3$. Then $\chi(H-e)\leq2$ exactly when some 2-coloring of $H$ makes $e$ its only monochromatic edge.

Rewritten proof

Suppose first that $H-e$ has a proper 2-coloring. The same coloring of $V(H)$ cannot be proper for $H$, since $\chi(H)\geq3$. Every edge other than $e$ is non-monochromatic, so $e$ must be its unique monochromatic edge.

Conversely, given a 2-coloring of $H$ whose only monochromatic edge is $e$, no edge of $H-e$ is monochromatic, so the same coloring is a proper 2-coloring of $H-e$.