Combinatorics of experimental design
Combinatorics of experimental design
Non-trivial t-designs without repeated blocks exist for all t
Discrete Mathematics
A completion of Lu's determination of the spectrum of large sets of disjoint Steiner Triple systems
Journal of Combinatorial Theory Series A
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We study partitions of the set of all {v}\choose{3} triples chosen from a v-set intopairwise disjoint planes with three points per line. Our partitions may contain copies of PG(2,2) only (Fano partitions) or copies of AG(2, 3) only (affine partitions)or copies of some planes of each type (mixed partitions).We find necessary conditions for Fano or affine partitions to exist. Such partitions are already known in severalcases: Fano partitions for v = 8 and affine partitions for v = 9 or 10. We constructsuch partitions for several sporadic orders, namely, Fano partitions for v = 14, 16, 22, 23, 28, andan affine partition for v = 18. Using these as starter partitions, we prove that Fano partitionsexist for v = 7n + 1, 13n + 1,27n + 1, and affine partitions for v = 8n + 1,9n + 1, 17n + 1. In particular, both Fano and affine partitionsexist for v = 36n + 1. Using properties of 3-wise balanced designs, weextend these results to show that affine partitions also exist for v = 32n.Similarly, mixed partitions are shown to exist for v = 8n,9n, 11n + 1.