Some results on the existence of large sets of t‐designs

A set of trivial necessary conditions for the existence of a large set of t ‐designs, LS [N]( t,k, ν), is $N\big | {{\nu \hskip -3.1 \nu}-i \choose k-i}$ for i  = 0,…, t . There are two conjectures due to Hartman and Khosrovshahi which state that the trivial necessary conditions are sufficient in the cases N  = 2 and 3, respectively. Ajoodani‐Namini has established the truth of Hartman's conjecture for t  = 2. Apart from this celebrated result, we know the correctness of the conjectures for a few small values of k , when N  = 2 and t  ≤ 6, and also when N  = 3 and t  ≤ 4. In this article, we show that similar results can be obtained for infinitely many values of k . © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 144–151, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10027