Large sets of disjoint packings on 6k+5 points

Abstract

A ð2; 3Þ-packing on X is a pair ðX ; AÞ; where A 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. Its leave is a graph ðX ; EÞ such that E consists of all the pairs which do not appear in any block of A: For a ð6k þ 5Þ-set X a large set of maximum packing, denoted by LMPð6k þ 5Þ; is a set of 6k þ 1 disjoint ð2; 3Þ-packings on X with a cycle of length four as their common leave. Schellenberg and Stinson (J. Combin. Math. Combin. Comput. 5 (1989) 143) first introduced such a large set problem and used it to construct perfect threshold schemes. In this paper, we show that an LMPð6k þ 5Þ exists for any positive integer k: This complete solution is based on the known existence result of Sð3; 4; vÞs by Hanani and that of 1-fan Sð3; 4; vÞs and Sð3; f4; 5; 6g; vÞs by the second author. Partitionable candelabra system also plays an important role together with two special known LMPð6k þ 5Þs for k ¼ 1; 2: r

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