Independence densities of hypergraphs

TL;DR

This paper introduces the concept of independence density in hypergraphs, proving rationality for uniform hypergraphs and the ability to attain any real value in [0,1] for non-uniform hypergraphs, extending the concept via polynomials.

Contribution

It establishes the rationality of independence densities for uniform hypergraphs and characterizes possible densities for non-uniform hypergraphs, extending the notion through independence polynomials.

Findings

Independence density is always rational for k-uniform hypergraphs.

Non-uniform hypergraphs can have independence densities anywhere in [0,1].

Extension of independence density via independence polynomials.

Abstract

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence polynomials.