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Counting Maximal Antichains and Independent Sets

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Abstract

Answering several questions of Duffus, Frankl and Rödl, we give asymptotics for the logarithms of (i) the number of maximal antichains in the n-dimensional Boolean algebra and (ii) the numbers of maximal independent sets in the covering graph of the n-dimensional hypercube and certain natural subgraphs thereof. The results in (ii) are implied by more general upper bounds on the numbers of maximal independent sets in regular and biregular graphs. We also mention some stronger possibilities involving actual rather than logarithmic asymptotics.

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References

  1. Chung, F.R.K., Frankl, P., Graham, R., Shearer, J.B.: Some intersection theorems for ordered sets and graphs. J. Comb. Theory Ser. A 48, 23–37 (1986)

    Article  MathSciNet  Google Scholar 

  2. Diestel, R.: Graph Theory. Springer, New York (2000)

    Google Scholar 

  3. Duffus, D., Frankl, P., Rödl, V.: Maximal independent sets in the covering graph of the cube. Discrete Appl. Math. doi:10.1016/j.dam.2010.09.003 (2010)

    Google Scholar 

  4. Duffus, D., Frankl, P., Rödl, V.: Maximal independent sets in bipartite graphs obtained from Boolean lattices. Eur. J. Comb. 32, 1–9 (2011)

    Article  MATH  Google Scholar 

  5. Engel, K.: Sperner Theory. Cambridge Univ. Pr., Cambridge (1997)

    Book  MATH  Google Scholar 

  6. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006)

    Google Scholar 

  7. Galvin, D.: An upper bound for the number of independent sets in regular graphs. Discrete Math. 309, 6635–6640 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hujter, M., Tuza, Z.: The number of maximal independent sets in triangle-free graphs. SIAM J. Discrete Math. 6, 284–288 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kahn, J.: An entropy approach to the hard-core model on bipartite graphs. Comb. Probab. Comput. 10, 219–237 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Kleitman, D.: On Dedekind’s problem: the number of isotone Boolean functions. Proc. Am. Math. Soc. 21, 677–682 (1969)

    MathSciNet  MATH  Google Scholar 

  11. McEliece, R.J.: The Theory of Information and Coding. Addison-Wesley, London (1977)

    MATH  Google Scholar 

  12. Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. 3, 23–28 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sapozhenko, A.A.: The number of independent sets in graphs. Moscow Univ. Math. Bull. 62, 116–118 (2007)

    Article  Google Scholar 

  14. Zhao, Y.: The number of independent sets in a regular graph. Comb. Probab. Comput. 19, 315–320 (2010)

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    L. Ilinca

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    J. Kahn

Authors
  1. L. Ilinca
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  2. J. Kahn
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Correspondence to J. Kahn.

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Supported by NSF grant DMS0701175

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Ilinca, L., Kahn, J. Counting Maximal Antichains and Independent Sets. Order 30, 427–435 (2013). https://doi.org/10.1007/s11083-012-9253-5

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  • DOI: https://doi.org/10.1007/s11083-012-9253-5

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