On the number of A-transversals in hypergraphs

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 H H and a non-empty set A of non-negative integers, we say that a set S is an A - transversal of H H if for any hyperedge H of H H , we have |H∩ S| ∈ A | H ∩ S | ∈ A . Independent sets are \0,1, ,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=\0,1, ,a\ A = { 0 , 1 , ⋯ , a } or A being the set of odd integers or the set of even integers.