Strongly minimal edge cover

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A hypergraph is $H=(V,E)$ with $E$ a family of finite subsets of a possibly infinite set $V$. There are no isolated vertices if $\bigcup E=V$. An edge cover is a subfamily $C\subseteq E$ with $\bigcup C=V$. It is strongly minimal if for every edge cover $C'\subseteq E$ one has $|C\setminus C'|\leq|C'\setminus C|$, where the inequality compares cardinalities, including infinite ones.

Statement recorded in the supplied source.

Does every such hypergraph admit a strongly minimal edge cover? This recorded finite-edge question has a negative answer in the cited source: the hypergraph consisting of all finite subsets of a countably infinite vertex set is a counterexample. Mere finiteness of each edge must be distinguished from a uniform upper bound on edge sizes. [S2]

Short English statement

Must a hypergraph with finite edges and no isolated vertices have an edge cover that cannot be improved by replacing more edges with fewer? The source gives a counterexample when edge sizes are unbounded.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3000085 (AMR-029-0085). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ron Aharoni, Strongly Maximal Matchings and Strongly Minimal Covers (2022), Definition 1.2 and section 2, preceding Conjecture 2.1, pp. 1--2. https://arxiv.org/abs/2206.02576.