On The Number Of Independent Sets In Uniform, Regular, Linear Hypergraphs
EUROPEAN JOURNAL OF COMBINATORICS(2022)
摘要
We study the problems of bounding the number weak and strong independent sets in r-uniform, d-regular, n-vertex linear hyper -graphs 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). (c) 2021 Published by Elsevier Ltd.
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关键词
independent sets
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