# Independent sets of a given size and structure in the hypercube

@inproceedings{Jenssen2021IndependentSO, title={Independent sets of a given size and structure in the hypercube}, author={Matthew Jenssen and Will Perkins and Aditya Potukuchi}, year={2021} }

Abstract. We determine the asymptotics of the number of independent sets of size ⌊β2⌋ in the discrete hypercube Qd = {0, 1} for any fixed β ∈ [0, 1] as d → ∞, extending a result of Galvin for β ∈ [1−1/ √ 2, 1]. Moreover, we prove a multivariate local central limit theorem for structural features of independent sets in Qd drawn according to the hard core model at any fixed fugacity λ > 0. In proving these results we develop several general tools for performing combinatorial enumeration using… Expand

#### 3 Citations

Approximate counting and sampling via local central limit theorems

- Computer Science, Mathematics
- ArXiv
- 2021

A new local central limit theorem for the hard-core model that applies to all fugacities below λc(∆), the uniqueness threshold on the infinite ∆-regular tree is proved. Expand

Enumerating independent sets in Abelian Cayley graphs

- Computer Science, Mathematics
- ArXiv
- 2021

We show that any connected Cayley graph Γ on an Abelian group of order 2n and degree Ω̃(log n) has at most 2(1+ o(1)) independent sets. This bound is tight up to to the o(1) term when Γ is bipartite.… Expand

Approximately counting independent sets in bipartite graphs via graph containers

- Computer Science, Mathematics
- ArXiv
- 2021

By implementing algorithmic versions of Sapozhenko’s graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies… Expand

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