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The Number of Maximal Independent Sets in the Hamming Cube

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Abstract

Let Qn be the n-dimensional Hamming cube and N = 2n. We prove that the number of maximal independent sets in Qn is asymptotically

$$2n{2^{N/4}},$$

as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and Rödl.

The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them “stability” results for maximal independent set counts and old and new results on isoperimetric behavior in Qn.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ, 08854-8019, USA

    Jeff Kahn

  2. Department of Mathematics, Stanford University, 450Jane Stanford Way, Building 380, Stanford, CA, 94305, USA

    Jinyoung Park

Authors
  1. Jeff Kahn
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  2. Jinyoung Park
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Correspondence to Jinyoung Park.

Additional information

The authors are supported by NSF Grant DMS1501962 and BSF Grant 2014290.

Jeff Kahn was supported by a Simons Fellowship.

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Kahn, J., Park, J. The Number of Maximal Independent Sets in the Hamming Cube. Combinatorica 42, 853–880 (2022). https://doi.org/10.1007/s00493-021-4729-9

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  • DOI: https://doi.org/10.1007/s00493-021-4729-9

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