Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least
${\exp\biggl({1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t} \biggl(\frac{1}{2}\ln t-1\biggr)\biggr)}$
independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as
n → ∞, every triangle-free graph on
n vertices has at least
${e^{(c_1-o(1)) \sqrt{n} \ln n}}$
independent sets, where
$c_1 = \sqrt{\ln 2}/4 = 0.208138 \ldots$
. 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
$c_2 = 2\sqrt{\ln 2} = 1.665109 \ldots$
. This disproves a conjecture from [8].
Let H be a (k+1)-uniform linear hypergraph with n vertices and average degree t. We also show that there exists a constant ck such that the number of independent sets in H is at least
${\exp\biggl({c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}\biggr})}.$
This is tight apart from the constant
ck and generalizes a result of Duke, Lefmann and Rödl [9], which guarantees the existence of an independent set of size
$\Omega\biggl(\frac{n}{t^{1/k}} \ln^{1/k}t\biggr).$
Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs.