I'd like to characterize the not-matchings of an hypergraph $H$, or more exactly its complements.
A not-matching is a set of edges containing at least $2$ not-disjoint edges. If $H$ is a graph this means it contains a path with 3 vertices (in the hypergraph case this just means it contains two adjacent edges, one of which might be contained in the other). The not-matchings are determined by the minimal ones, so their complements consist of the complements of the paths with $3$ vertices in the graph. As it's said here in page 97, when $H$ is a complete graph with $n$ vértices these complements can be made disconnected by removing $n-2$ vertices, i.e. the complements of the not matchings are the not $n-2$-connected graphs on $n$ vertices (actually, the ones which, ignoring their isolated vertices, are not $n-2$-connected if I'm not mistaken, since they are determined by the sets of edges).
Is there a similar characterization when $H$ is not the complete graph, or when $H$ is a general hypergraph?
For those familiar with simplicial stuff, I'm trying to compute the Alexander dual of the matching complex of a general graph/hypergraph.