Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Pra{\l}at about independence densities of graphs. Furthermore, we show that for any $k$, the set of independence densities of hypergraphs with hyperedges of size at most $k$ is closed and contains no infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page

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 $$e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)}$$ 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