Counterexamples to conjectures on strong maximality and minimality

TL;DR

The paper constructs counterexamples to natural conjectures on strong maximality and strong minimality for matchings, edge-covers, and colourings in infinite graphs and hypergraphs, including a $3$-uniform hypergraph with no strongly maximal matching and another with no strongly minimal edge-cover, from which a graph with no strongly minimal colouring is derived. A central device is a gadget $G_e$ that simulates large edges using only $2$- and $3$-edges, enabling reductions from unbounded-edge counterexamples $H_1^*$ and $H_2^*$ to bounded-edge constructions. The authors also address a Tardos-type question, providing a hypergraph $H_3$ with the property that for every matching $M$ there exists $M'$ with $|M\setminus M'|=1$ and $|M'\setminus M|=2$, thereby negating the proposed positive answer. Additional sections connect these results to flag complexes and colouring, and the concluding remarks outline open problems and directions, including fractional analogues. The work significantly refines the landscape of maximality/minimality phenomena in infinite combinatorics and negates several natural infinite-counterpart conjectures.

Abstract

We provide counterexamples to several conjectures concerning strongly maximal and strongly minimal structures in infinite graphs and hypergraphs. In particular, we construct 3-uniform hypergraphs without strongly maximal matchings and without strongly minimal covers, and from our construction for covers we build a graph with no strongly minimal colouring. We also consider several refinements of these problems. Our results resolve conjectures and questions of Aharoni; Aharoni and Berger; Aharoni, Berger, Georgakopoulos, and Sprüssel; Aharoni and Korman; and Tardos.

Figures (2)

• Figure 1: The gadget $G_e$ on the edge $e=\{v_1,\dots,v_5\}$, with inner edges marked in red, and outer edges in grey and black.
• Figure 2: A matching in the hypergraph $H_3$. The edge filled in red can be replaced by the edges drawn in dashed blue.

Theorems & Definitions (21)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 3.1
• proof