Counting Independent Sets in Hypergraphs

Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least

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

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.