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 e(1−n −1/12) 1 2 n t ln t( 1 2 ln t−1) 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(c1−o(1)) √ n lnn independent sets, where c1 = √ 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(c2+o(1)) √ n lnn independent sets, where c2 = 1 + ln 2 = 1.693147... 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