Complete subgraphs of the coprime hypergraph of integers II: structural properties

Abstract

We continue to investigate structural properties of complete subgraphs of the coprime hypergraph of integers on n vertices , which has vertex set and hyperedge set . It turns out that any maximal complete subgraph G of has only vertices with at most three prime divisors for sufficiently large n. Further, we show that a vertex a of G with three prime divisors is squarefree and satisfies . We also give upper bounds for the number of prime divisors of vertices of G in case of arbitrary n.

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Fig. 1

Notes

  1. 1.

    This operation has already been used, although not properly introduced, in [1].

References

  1. 1.

    de Wiljes, J.-H.: Complete subgraphs of the coprime hypergraph of integers I: introduction and bounds. Eur. J. Math. 3(2), 379–386 (2017)

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    Koshy, T.: Catalan Numbers with Applications. Oxford University Press, Oxford (2008)

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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 II: structural properties. European Journal of Mathematics 4, 676–686 (2018). https://doi.org/10.1007/s40879-017-0176-y

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Keywords

  • Hypergraphs on integers
  • Clique
  • Catalan numbers

Mathematics Subject Classification

  • 11B75
  • 05C69
  • 05C65