Proposition 4.6: vertex-criticality of the construction

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Updated Oct 8

Source. Ruiliang Li, On an Erdős--Lovász problem: 3-critical 3-graphs of minimum degree 7, arXiv:2512.24850v1 (31 December 2025), Proposition 4.6 and proof, printed p. 10, with certificates in Appendix B, Table 2, printed p. 12 (PDF pp. 10 and 12).

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 every $v\in V(H)$ , the vertex-deleted hypergraph $H-v$ is 2-colorable.

Rewritten proof

For each deleted vertex $v$ , color the listed vertices blue and every other remaining vertex red:

$v$	blue class	$v$	blue class	$v$	blue class
$1$	$\{2,3,4,5\}$	$4$	$\{1,2,5,6\}$	$7$	$\{1,2,4,5\}$
$2$	$\{1,3,4,5\}$	$5$	$\{1,2,4,7\}$	$8$	$\{1,2,4,7\}$
$3$	$\{1,2,4,5\}$	$6$	$\{1,2,4,5\}$	$9$	$\{1,2,6,8\}$

For each column pair, direct comparison with the 22-edge list shows that every edge avoiding $v$ meets both color classes. The indicated coloring is therefore proper on $H-v$ . These nine comparisons are also checked by evidence/verify_e0834_hypergraph.py.

Linked from (5)

Problem 834 set_systems/li_2025_erdos_lovasz_problem_3_critical Li's proof review and finite checks Review of Li's two critical-hypergraph proofs Theorem 4.1: the nine-vertex construction

Graph