论文标题
$ a+a+b \ ne c $解决方案在阿贝尔小组中以及计算超图中的独立集的应用
Number of $A+B\ne C$ solutions in abelian groups and application to counting independent sets in hypergraphs
论文作者
论文摘要
本文处理了添加剂组合学的问题。令$ {\ Mathbf G} $为有限的Abelian订单$ n $。我们证明了子集的数量$ a，b，b，c \ subset {\ mathbf g} $，以便对于$ x \ in B $中的任何$ x \ in b $ in b $中的任何$ x \ in b $ in b $ in和$ z \ in c $ y in c $一个has $ x+y \ y \ y \ ne z $ equals $ quall 3 \ cdot 4^n+n3^{n+1}+o（（（3-c _*）^n）$$对于某些绝对常数$ c _*> 0 $。这为特殊的3均匀线性超图中的独立集数量提供了严格的估计，并支持了自然猜想，涉及$ n $顶点的此类超图中最大可能的独立集。
The paper deals with a problem of Additive Combinatorics. Let ${\mathbf G}$ be a finite abelian group of order $N$. We prove that the number of subset triples $A,B,C\subset {\mathbf G}$ such that for any $x\in A$, $y\in B$ and $z\in C$ one has $x+y\ne z$ equals $$ 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.
