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Partition of Triples of Order 6k+5 into 6k+3 Optimal Packings and One Packing of Size 8k+4

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Abstract.

A (2,3)-packing on X is a pair (X,), where is a set of 3-subsets (called blocks) of X, such that any pair of distinct points from X occurs together in at most one block. For a (6k+5)-set X, an optimal partition of triples (denoted by OPT(6k+5)) is a set of 6k+3 optimal (2,3)-packings and a (2,3)-packing of size 8k+4 on X. Etzion conjectured that there exists an OPT(6k+5) for any positive integer k. In this paper, we construct such a system for any k≥1. This complete solution is based on the known existence results of S(3,4,v)s by Hanani and that of special S(3,{4,6},6m)s by Mills. Partitionable candelabra systems also play an important role together with an OPT(11) and a holey OPT(11).

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Authors and Affiliations

  1. Department of Mathematics, Suzhou University, Suzhou, 215006, China

    Lijun Ji

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  1. Lijun Ji
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Correspondence to Lijun Ji.

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Research supported by Natural Science Foundation of Universities of Jiangsu Province under Grant 05KJB110111

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Ji, L. Partition of Triples of Order 6k+5 into 6k+3 Optimal Packings and One Packing of Size 8k+4. Graphs and Combinatorics 22, 251–260 (2006). https://doi.org/10.1007/s00373-005-0632-1

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