On the number of A-transversals in hypergraphs

Abstract

AbstractA 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 $${{\mathcal {H}}}$$ H and a non-empty set A of non-negative integers, we say that a set S is an A-transversal of $${{\mathcal {H}}}$$ H if for any hyperedge H of $${{\mathcal {H}}}$$ H , we have $$|H\cap S| \in A$$ | H ∩ S | ∈ A . Independent sets are $$\{0,1,\dots ,r{-}1\}$$ { 0 , 1 , ⋯ , r - 1 } -transversals, while strongly independent sets are $$\{0,1\}$$ { 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 = { a } , $$A=\{0,1,\dots ,a\}$$ A = { 0 , 1 , ⋯ , a } or A being the set of odd integers or the set of even integers.