Number of [formula omitted] solutions in abelian groups and application to counting independent sets in hypergraphs

Abstract

The paper deals with a problem of Additive Combinatorics. Let G be a finite abelian group of order N. We prove that the number of subset triples A,B,C⊂G such that for any x∈A, y∈B and z∈C one has x+y≠z equals 3⋅4N+N3N+1+O((3−c∗)N) for some absolute constant c∗>0. This provides a tight estimate for the number of independent sets in a special 3-uniform linear hypergraph and disproves the conjecture of Cohen et al. (0000). concerning the maximal possible number of independent sets in such hypergraphs on n vertices.