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arXiv · 2012.13433

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

Aliaksei Semchankau·Dmitry Shabanov·Ilya Shkredov
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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 $$ 3\cdot 4^N+N3^{N+1} + O((3-c_*)^N) $$ 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 gives a support for the natural conjecture concerning the maximal possible number of independent sets in such hypergraphs on $n$ vertices.

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Aliaksei Semchankau, Dmitry Shabanov, Ilya Shkredov. 2020-12-24. Number of $A+B\ne C$ solutions in abelian groups and application to counting independent sets in hypergraphs. https://arxiv.org/abs/2012.13433

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