On the number of \(A\)-transversals in hypergraphs (Q6607826)

A set \(S\) of vertices in a hypergraph is called strongly independent if every hyperedge shares at most one vertex with \(S\). In this paper, the authors prove a sharp result for the number of maximal strongly independent sets in a 3-uniform hypergraph analogous to the Moon-Moser theorem. Given an \(r\)-uniform hypergraph \(\mathcal{H}\) and a non-empty set \(A\) of non-negative integers, a set \(S\) is called an \(A\)-transversal of \(\mathcal{H}\) if for any hyperedge \(H\) of \(\mathcal{H}\), we have \(\vert H \cap S\vert \in A\). Independent sets are \(\{0, 1, \ldots, r-1\}\)-transversals, while strongly independent sets are \(\{0, 1\}\)-transversals. Note that for some sets \(A\), there may exist hypergraphs without any \(A\)-transversals. The authors study the maximum number of \(A\)-transversals for every \(A\), with emphasis on the more natural sets, \(A = \{a\}, A = \{0, 1,\ldots, a\}\) or \(A\) being the set of odd integers or the set of even integers.