Counterexamples to Conjectures on Strong Maximality and Minimality

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.