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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 4.2 and proof, printed p. 8 (PDF p. 8).

Setup. The construction (5) of Theorem 4.1.

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

For the hypergraph $H$ of Theorem 4.1,

$$d_H(1)=10,\qquad d_H(v)=7\quad(2\leq v\leq9).$$

In particular, $\delta(H)=7$.

Rewritten proof

Reading the edge list and grouping the edges by vertex gives

$$\begin{array}{c|l} 1&123,129,138,146,148,149,157,158,159,167\\ 2&123,129,236,237,249,259,267\\ 3&123,138,236,237,348,358,367\\ 4&146,148,149,249,348,468,469\\ 5&157,158,159,259,358,578,579\\ 6&146,167,236,267,367,468,469\\ 7&157,167,237,267,367,578,579\\ 8&138,148,158,348,358,468,578\\ 9&129,149,159,249,259,469,579. \end{array}$$

The first row contains ten edges and every other row contains seven, proving the claim. As a consistency check, the degree sum is $10+8\cdot7=66$, equal to three times the 22 edges.