Support sizes of threefold resolvable triple systems

Let SS R ( v , 3) denote the set of all integer b * such that there exists a RTS ( v , 3) with b * distinct triples. In this paper, we determine the set SS R ( v , 3) for v ≡ 3 (mod 6) and v ≥ 3 with only five undecided cases. We establish that SS R ( v , 3) = P ( v , 3) for v ≡ 3 (mod 6), v ≥ 21 and v ≠ 33, 39 where P ( v , 3) = { m v , m v + 4, m v + 6, m v + 7, …, 3 m v } and m v , = v ( v − 1)/6. As a by‐product, we remove the last two undecided cases for the intersection numbers of Kirkman triple system of order 27, this improves the known result provided in [2]. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 275–289, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10037