  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).
