On the Number of Independent Sets in Uniform, Regular, Linear Hypergraphs

TL;DR

This paper derives tight bounds on the number of weak and strong independent sets in uniform, regular, linear hypergraphs, advancing understanding of hypergraph independence structure with new bounds and constructions.

Contribution

It provides new tight bounds for independent sets in hypergraphs, including explicit constructions and improved results for specific cases, using the occupancy method.

Findings

Upper bounds for weak independent sets are tight up to first order for all fixed r≥3.

Upper bounds for strong independent sets are tight up to second order for r=3.

Explicit construction of a 3-uniform, d-regular, linear hypergraph with no cross-edges.

Abstract

We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) $r\ge 3$, with $d$ and $n$ going to infinity. In the case of strong independent sets, for $r=3$, we provide an upper bound that is tight up to the second-order term, improving on a result of Ordentlich-Roth (2004). The tightness in the strong independent set case is established by an explicit construction of a $3$-uniform, $d$-regular, cross-edge free, linear hypergraph on $n$ vertices which could be of interest in other contexts. We leave open the general case(s) with some conjectures. Our proofs use the occupancy method introduced by Davies, Jenssen, Perkins, and Roberts (2017).