Invertible Families of Sets of Bounded Degree

Let H = (H; V) be a hypergraph with edge set H and vertex set V. Then H is invertible ii there exists a permutation of V such that for all E 2 H, (E) \ E = ;. H is invertibility critical if H is not invertible but every hypergraph obtained by removing an edge from H is invertible. The degree of H is d if fE 2 Hjx 2 Eg d for each x 2 V. Let i(d) be the maximum number of edges of an invertibility critical hypergraph of degree d. Theorem: i(d) (d ? 1) ? 2d?1 d + 1. The proof of this result leads to the following covering problem on graphs: Let G be a graph. A family H 2 V (G) is an edge cover of G ii for every edge e of G, there is an E 2 H which includes e. H is a minimal edge cover of G ii for H 0 H, H 0 is not an edge cover of G. Let b(d) (c(d)) be the maximum cardinality of a minimal edge cover H of a complete bipartite graph (complete graph) where H has degree d. Theorem: c(d) i(d) b(d) c(d + 1) and 3 2 d?1 ? 2 b(d) (d ? 1) ? 2d?1 d + 1. The proof of this result uses Sperner theory. The bounds b(d) also arise as bounds on the maximum number of elements in the union of minimal covers of families of sets.