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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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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DOI: https://doi.org/10.1007/s00493-011-2639-y