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General Independence Sets in Random Strongly Sparse Hypergraphs

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Abstract

We analyze the asymptotic behavior of the j-independence number of a random k-uniform hypergraph H(n, k, p) in the binomial model. We prove that in the strongly sparse case, i.e., where \(p = c/\left( \begin{gathered} n - 1 \hfill \\ k - 1 \hfill \\ \end{gathered} \right)\) for a positive constant 0 < c ≤ 1/(k − 1), there exists a constant γ(k, j, c) > 0 such that the j-independence number α j (H(n, k, p)) obeys the law of large numbers \(\frac{{{\alpha _j}\left( {H\left( {n,k,p} \right)} \right)}}{n}\xrightarrow{P}\gamma \left( {k,j,c} \right)asn \to + \infty \) Moreover, we explicitly present γ(k, j, c) as a function of a solution of some transcendental equation.

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Authors and Affiliations

  1. Department of Probability Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    A. S. Semenov & D. A. Shabanov

  2. Chair of Discrete Mathematics, Department of Innovation and High Technology, Moscow Institute of Physics and Technology (State University), Moscow, Russia

    A. S. Semenov

  3. Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology (State University), Moscow, Russia

    D. A. Shabanov

Authors
  1. A. S. Semenov
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  2. D. A. Shabanov
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Correspondence to A. S. Semenov.

Additional information

Original Russian Text © A.S. Semenov, D.A. Shabanov, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 1, pp. 63–77.

Supported in part by the Russian Foundation for Basic Research, project no. 15-01-03530-a, and President of the Russian Federation Grant no. MD-5650.2016.1.

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Semenov, A.S., Shabanov, D.A. General Independence Sets in Random Strongly Sparse Hypergraphs. Probl Inf Transm 54, 56–69 (2018). https://doi.org/10.1134/S0032946018010052

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  • DOI: https://doi.org/10.1134/S0032946018010052

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