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

TL;DR

This paper proves that positive independence densities of countable hypergraphs with bounded hyperedge size are achieved by finite hypergraphs, answering a key question and revealing structural properties of these densities.

Contribution

It demonstrates that such densities are always realized by finite hypergraphs and establishes the closedness and absence of infinite increasing sequences in their density sets.

Findings

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

The set of independence densities for hypergraphs with bounded hyperedge size is closed.

There are no infinite increasing sequences in these density sets.

Abstract

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.