# Strongly maximal matchings, strongly minimal vertex covers

For a hypergraph $H=(V,E)$, we call $M \subseteq E$ a matching if the elements of $M$ are pairwise disjoint, and we call $C\subseteq V$ a vertex cover if it meets every hyperedge. A matching $M$ is strongly maximal if $\left|M'\setminus M\right|\leq \left|M\setminus M'\right|$ holds for any matching $M'$. A vertex cover $C$ is strongly minimal if $\left|C\setminus C'\right|\leq \left|C'\setminus C\right|$ holds for any cover $C'$.

Is it true that if all the hyperedges of a hypergraph $H$ have size at most $k$ for some $k\in \mathbb{N}$, then $H$ admits a strongly maximal matching and a strongly minimal cover?

## Remarks

In the case of graphs the existence of a strongly maximal matching is known to be true ([1] p. 16. Theorem 5.6). The existence of strongly maximal matching conjectured originally under the assumption that the hyperedges are finite but it turned out to be false [2].

## References

1. R. Diestel and C.St.J.A. Nash-Williams, Directions in infinite graph theory and combinatorics, North-Holland, 1992. author link
2. Ahlswede, Rudolf, and Levon H. Khachatrian. "A counterexample to Aharoni's strongly maximal matching conjecture." Discrete Mathematics 149.1 (1996): 289.