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Bounds on Threshold Probabilities for Coloring Properties of Random Hypergraphs

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Abstract

We study the threshold probability for the property of existence of a special-form \(r\)⁠-⁠coloring for a random \(k\)⁠-⁠uniform hypergraph in the \(H(n,k,p)\) binomial model. A parametric set of \(j\)⁠-⁠chromatic numbers of a random hypergraph is considered. A coloring of hypergraph vertices is said to be \(j\)⁠-⁠proper if every edge in it contains no more than \(j\) vertices of each color. We analyze the question of finding the sharp threshold probability of existence of a \(j\)⁠-⁠proper \(r\)⁠-⁠coloring for \(H(n,k,p)\). Using the second moment method, we obtain rather tight bounds for this probability provided that \(k\) and \(j\) are large as compared to \(r\).

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Funding

The research of A. Semenov was supported by the Russian Foundation for Basic Research, project no. 18-31-00348. The research of D. Shabanov was supported by the Russian Foundation for Basic Research, project no. 20-31-70039, and partially supported by the President of the Russian Federation grant no. MD-1562.2020.1.

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Authors and Affiliations

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

    A. S. Semenov

  2. Laboratory of Combinatorial and Geometric Structures, Moscow Institute of Physics and Technology (State University), Moscow, Russia

    D. A. Shabanov

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

    D. A. Shabanov

  4. Faculty of Computer Science, Higher School of Economics—National Research University, Moscow, Russia

    D. A. Shabanov

Authors
  1. A. S. Semenov
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  2. D. A. Shabanov
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Additional information

Translated from Problemy Peredachi Informatsii, 2022, Vol. 58, No. 1, pp. 80–111 https://doi.org/10.31857/S0555292322010053.

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Semenov, A., Shabanov, D. Bounds on Threshold Probabilities for Coloring Properties of Random Hypergraphs. Probl Inf Transm 58, 72–101 (2022). https://doi.org/10.1134/S0032946022010057

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