Number of $A+B\ne C$ solutions in abelian groups and application to counting independent sets in hypergraphs

Abstract

The paper deals with a problem of Additive Combinatorics. Let $\mathbf{G}$ be a finite abelian group of order $N$. We prove that the number of subset triples $A,B,C\subset \mathbf{G}$ such that for any $x\in A$, $y\in B$ and $z\in C$ one has $x+y\ne z$ equals $$3\cdot 4^N+N{3}^{N+1}+O((3-c_{\ast }{)}^N)$$ for some absolute constant $c_{\ast }>0$. This provides a tight estimate for the number of independent sets in a special 3-uniform linear hypergraph and gives a support for the natural conjecture concerning the maximal possible number of independent sets in such hypergraphs on $n$ vertices.

15 pages

References in corpus (1)

• A note on Pollard's Theorem