On an Erd\H{o}s--Lov'asz problem: 3-critical 3-graphs of minimum degree 7

TL;DR

This paper investigates Erdős and Lovász's question about the existence of 3-critical 3-uniform hypergraphs with minimum degree 7, providing bounds and explicit examples under two different notions of criticality.

Contribution

The paper resolves the question for both interpretations of 3-criticality, establishing degree bounds and constructing explicit examples demonstrating the existence of such hypergraphs.

Findings

For the transversal interpretation, maximum edges are 10, with minimum degree at most 6, achieved by the complete 3-graph on 5 vertices.

For the chromatic interpretation, an explicit 9-vertex hypergraph with minimum degree 7 is constructed, which becomes 2-colorable upon removal of any edge or vertex.

Abstract

Erd\H{o}s and Lov'asz asked whether there exists a "3-critical" 3-uniform hypergraph in which every vertex has degree at least 7. The original formulation does not specify what 3-critical means, and two non-equivalent notions have appeared in the literature and in later discussions of the problem. In this paper we resolve the question under both interpretations. For the transversal interpretation (criticality with respect to the transversal number), we prove that a 3-uniform hypergraph $H$ with $\tau(H)=3$ and $\tau(H-e)=2$ for every edge $e$ has at most 10 edges; in particular, $\delta(H)\le 6$, and this bound is sharp, witnessed by the complete 3-graph $K^{(3)}_5$. For the chromatic interpretation (criticality with respect to weak vertex-colourings), we give an explicit 3-uniform hypergraph on 9 vertices with $\chi(H)=3$ and minimum degree $\delta(H)=7$ such that deleting any single edge or any single vertex makes it 2-colourable. The criticality of the example is certified by explicit witness 2-colourings listed in the appendices, together with a short verification script.