Abstract
A (2,3)-packing on X is a pair (X, A) , 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 ( X, E) such that E consists of all the pairs which do not appear in any block of A . For a (6 k+5)-set X a large set of maximum packing, denoted by LMP(6 k+5), is a set of 6 k+1 disjoint (2,3)-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 LMP(6 k+5) exists for any positive integer k. This complete solution is based on the known existence result of S(3,4, v)s by Hanani and that of 1-fan S(3,4, v)s and S(3,{4,5,6}, v)s by the second author. Partitionable candelabra system also plays an important role together with two special known LMP(6 k+5)s for k=1,2.
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