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).

我们研究了数个弱独立集和强独立集的边界问题 $r$-制服， $d$-常规的， $n$- 没有交叉边的顶点线性超图。在弱独立集的情况下，我们提供了一个上限，该上限紧至所有（固定）的一阶项$r\ge 3$， 和 $d$ 和 $n$走向无限。在强独立集的情况下，对于$r=3$，我们提供了一个紧至二阶项的上限，改进了 Ordentlich-Roth (2004) 的结果。强独立集情况下的紧密性是通过 3-uniform 的显式构造建立的，$d$- 规则的、无交叉边的、线性超图 $n$在其他上下文中可能感兴趣的顶点。我们留下一些猜想的一般情况。我们的证明使用 Davies、Jenssen、Perkins 和 Roberts (2017) 引入的占用方法。