论文标题
在统一，正常的线性超图中的独立集数量上
On the Number of Independent Sets in Uniform, Regular, Linear Hypergraphs
论文作者
论文摘要
我们研究了以$ r $统一，$ d $ rongular，$ n $ -vertex线性超图，没有交叉边缘的限制数字弱且强大的独立设置的问题。在弱独立集的情况下，我们提供了一个上限，该界限紧密到所有（固定）$ r \ ge 3 $的第一阶任期，其中$ d $和$ n $将用于无限。在强大的独立集中，对于$ r = 3 $，我们提供了一个上限，直到二阶任期都紧密，从而改善了Ordentlich-Roth（2004）的结果。强大的独立设定案例中的紧密度是通过明确构造$ 3 $均匀的，$ d $的$ d $ - 跨边缘，跨边缘的，线性超图在其他情况下可能引起的。我们将一般案例留下一些猜想。我们的证明使用戴维斯，詹森，珀金斯和罗伯茨（2017）引入的占用方法。
We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) $r\ge 3$, with $d$ and $n$ going to infinity. In the case of strong independent sets, for $r=3$, we provide an upper bound that is tight up to the second-order term, improving on a result of Ordentlich-Roth (2004). The tightness in the strong independent set case is established by an explicit construction of a $3$-uniform, $d$-regular, cross-edge free, linear hypergraph on $n$ vertices which could be of interest in other contexts. We leave open the general case(s) with some conjectures. Our proofs use the occupancy method introduced by Davies, Jenssen, Perkins, and Roberts (2017).
