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.
Bibliographical noteFunding Information:
Thanks to Alex Rosa for very helpful comments,a nd for assistancein tracking down the literature cited. Research of the first author is supportedb y NSERC Canada under grant number A0579. This researchw as begun while the second author was visiting Department of Mathematics, University of Toronto, and continuedw hile the first author visited IBM Israel. We expresst hanks to these institutions.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics