Complete subgraphs of the coprime hypergraph of integers I: introduction and bounds

Abstract

We introduce the uniform coprime hypergraph of integers \(\mathrm{CHI}_k\) which is the graph with vertex set \({\mathbb {Z}}\) and a -hyperedge exactly between every \(k+1\) elements of \({\mathbb {Z}}\) having greatest common divisor equal to 1. This generalizes the concept of coprime graphs. We obtain some basic properties of these graphs and give upper and lower bounds for the clique number of certain subgraphs of \(\mathrm{CHI}_k\).

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Notes

  1. 1.

    Actually they proved a stronger version where l does not have to be larger than m, although in that case we would not have such a nice graph theoretic interpretation.

  2. 2.

    Every tree T on n vertices is isomorphic to a subgraph of . This conjecture was proved in 2011 [10] for all sufficiently large n.

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Correspondence to Jan-Hendrik de Wiljes.

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de Wiljes, JH. Complete subgraphs of the coprime hypergraph of integers I: introduction and bounds. European Journal of Mathematics 3, 379–386 (2017). https://doi.org/10.1007/s40879-017-0137-5

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Keywords

  • Hypergraphs on integers
  • Clique number
  • Degree sequence

Mathematics Subject Classification

  • 11B75
  • 05C65
  • 05C69