On the number of independent sets in uniform, regular, linear hypergraphs. (January 2022)

• Record Type: Journal Article
• Authors: Cohen, Emma; Perkins, Will; Sarantis, Michail; Tetali, Prasad
• Is Part Of: European journal of combinatorics. Volume 99(2022)
• Publication Date: 2022-01
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2021.103401
• Languages: English
• ISSNs: 0195-6698

Abstract

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 ≥ 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).