Abstract
A lambda-design is a family B={B(1),B(2),..., B(v)} of subsets of x={1, 2, ..., v} such that vertical bar B(i)boolean AND B(j)vertical bar=lambda for all i not equal j and not all B(j) are of the same size. The only known example of lambda-designs (called type-1 designs) are those obtained from symmetric designs by a certain complementation procedure. Ryser Ii Algebra 10 (1968), 246-261] and Woodall [Proc London Math Soc 20 (1970), 669-687] independently conjectured that all lambda-designs are type-1. Let g=gcd(r-1, r*-1), where r and r* are the two replication numbers. Ionin and Shrikhande [J Combin Comput 22 (1996), 135-142; J Combin Theory Ser A 74 (1996), 100-114] showed that lambda-designs with g=1, 2, 3, 4 are type-1 and that the Ryser-Woodall conjecture is true for lambda-designs on p+1, 2p+1, 3p+1, 4p+1 points, where p is a prime. Hein and Ionin [Codes and Designs-Proceedings of Conference honoring Prof. D. K. Ray-Chaudhuri on the occasion of his 65th birthday, Ohio State University Mathematical Research Institute Publications, 10, Walter de Gruyter, Berlin, 2002, pp. 145-156] proved corresponding results for g=5 and Fiala [Codes and Designs-Proceedings of Conference honoring Prof. D. K. Ray-Chaudhuri on the occasion of his 65th birthday, Ohio State University Mathematical Research Institute Publications, 10, Walter de Gruyter, Berlin, 2002, pp. 109-124; Ars Combin 68 (2003), 17-32; Ars Combin, to appear] for g=6, 7, and 8. In this article, we consider lambda designs with exactly two block sizes. We show that in this case, the conjecture is true for g=9, 11, 12, 13, 15, 16, 17, 19, 20, 21, and for g = 10, 14, 18, 22 with v not equal 4 lambda-1. We also give two results on such lambda-designs on v=9p + 1 and 12p+1 points, where p is a prime. (C) 2010 Wiley Periodicals, Inc. J Combin Designs 19: 95-110, 2011