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|Paper IPM / M / 771||
A set of trivial necessary conditions for the existence of a large
set of t-designs, LS[N](t,k,v), is N|((v−i) || (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 the foregoing results are in fact true for
infinitely many values of k.
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