On the non-existence of pair covering designs with at least as many points as blocks

Combinatorica volume 31, pages507–528(2011)Cite this article

Abstract

We establish new lower bounds on the pair covering number C λ (υ,k) for infinitely many values of υ, k and λ, including infinitely many values of υ and k for λ=1. Here, C λ (υ,k) denotes the minimum number of k-subsets of a υ-set of points such that each pair of points occurs in at least λ of the k-subsets. We use these results to prove simple numerical conditions which are both necessary and sufficient for the existence of (K k − e)-designs with more points than blocks.

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Affiliations

  1. Department of Mathematics, The University of Queensland, Brisbane, QLD, 4072, Australia

    Darryn Bryant, Melinda Buchanan, Barbara Maenhaut & Victor Scharaschkin

  2. School of Mathematical Sciences, Monash University, Melbourne, VIC, 3800, Australia

    Daniel Horsley

Authors
  1. Darryn Bryant
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  2. Melinda Buchanan
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  3. Daniel Horsley
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  4. Barbara Maenhaut
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  5. Victor Scharaschkin
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Correspondence to Darryn Bryant.

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Bryant, D., Buchanan, M., Horsley, D. et al. On the non-existence of pair covering designs with at least as many points as blocks. Combinatorica 31, 507–528 (2011). https://doi.org/10.1007/s00493-011-2639-y

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Mathematics Subject Classification (2000)

  • 05B40
  • 05B30
  • 05C51
  • 05C70