Large sets of disjoint packings on 6k+5 points

摘要

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 (6k + 5)-set X a large set of maximum packing, denoted by LMP(6k + 5), is a set of 6k + 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(6k + 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(6k + 5)s for k = 1, 2. (C) 2004 Elsevier Inc. All rights reserved.