On the number of A-transversals in hypergraphs

Abstract

A set S of vertices in a hypergraph is strongly independent if every hyperedge shares at most one vertex with S. We 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, we say that a set S is an A-transversal of \({{\mathcal {H}}}\) if for any hyperedge H of \({{\mathcal {H}}}\), we have \(|H\cap S| \in A\). Independent sets are \(\{0,1,\dots ,r{-}1\}\)-transversals, while strongly independent sets are \(\{0,1\}\)-transversals. Note that for some sets A, there may exist hypergraphs without any A-transversals. We study the maximum number of A-transversals for every A, but we focus on the more natural sets, \(A=\{a\}\), \(A=\{0,1,\dots ,a\}\) or A being the set of odd integers or the set of even integers.