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Problems of Information Transmission

, Volume 54, Issue 1, pp 56–69 | Cite as

General Independence Sets in Random Strongly Sparse Hypergraphs

  • A. S. Semenov
  • D. A. Shabanov
Large Systems
  • 18 Downloads

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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Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.Department of Probability Theory, Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Chair of Discrete Mathematics, Department of Innovation and High TechnologyMoscow Institute of Physics and Technology (State University)MoscowRussia
  3. 3.Laboratory of Advanced Combinatorics and Network ApplicationsMoscow Institute of Physics and Technology (State University)MoscowRussia

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