  Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We
show that $G$ contains at least \[ e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t
(\frac{1}{2}\ln t-1)} \] independent sets. This improves a recent result of the
first and third authors \cite{countingind}. In particular, it implies that as
$n \to \infty$, every triangle-free graph on $n$ vertices has at least
$e^{(c_1-o(1)) \sqrt{n} \ln n}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 =
0.208138..$. Further, we show that for all $n$, there exists a triangle-free
graph with $n$ vertices which has at most $e^{(c_2+o(1))\sqrt{n}\ln n}$
independent sets, where $c_2 = 1+\ln 2 = 1.693147..$. This disproves a
conjecture from \cite{countingind}.
  Let $H$ be a $(k+1)$-uniform linear hypergraph with $n$ vertices and average
degree $t$. We also show that there exists a constant $c_k$ such that the
number of independent sets in $H$ is at least \[ e^{c_{k}
\frac{n}{t^{1/k}}\ln^{1+1/k}{t}}. \] This is tight apart from the constant
$c_k$ and generalizes a result of Duke, Lefmann, and R\"odl
\cite{uncrowdedrodl}, which guarantees the existence of an independent set of
size $\Omega(\frac{n}{t^{1/k}} \ln^{1/k}t)$. Both of our lower bounds follow
from a more general statement, which applies to hereditary properties of
hypergraphs.

Combinatorics;Discrete Mathematics