The possible intersection sizes for Steiner quadruple systems are examined. The determination of possible intersection sizes for v ≡ 4, 8 (mod 12), v ≥ 40, was recently completed by Lo Faro. For v ≡ 0 (mod 6), v ≥ 42, we solve completely the analogous intersection problem for threewise balanced designs with a spanning set of blocks of size 6, and blocks of size four otherwise. For v ≡ 2 (mod 12), v ≥ 38, we solve the intersection problem except when the intersection size is less than (v - 2)(v - 14)/6. For v ≡ 10 (mod 12), v ≥ 46, we solve the intersection problem except when the intersection size is less than (v - 10)/6. Using these results on intersection, we obtain substantial partial results on the possible support sizes of quadruple systems with λ = 2 and 3.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics