Quasi-symmetric triangle-free designs D with block intersection numbers 0 and y and with no three mutually disjoint blocks are studied. It is shown that the parameters of D are expressible in terms of only two parameters y and m, where m = k/y, k being the block size.
Baartmans and Shrikhande proved that 2~-<m ~<y + 1 and characterized the extremal values of m. An alternative characterization of the extremal cases and also an alternative proof of the bounds is obtained. It is conjectured that besides the extremal cases, there are only finitely many such designs. It is proved that in such designs if k is a prime power p", then p = 2 and D is a Hadamard design.
1. latroduetion
Quasi-symmetric designs have been the objects of much investigation in recent years. These are 2-designs D with precisely two block intersection numbers x and y; this paper restricts itself to x = 0. It is well-known that the block graph F of a quasi-symmetric design D is strongly regular and the block intersection number y divides the block size k (see [8], for example.) Particularly interesting quasisymmetric designs are those in which the complement of the block graph F is triangle-free, i.e., D has no three mutually disjoint blocks. We call such a quasi-symmetric design triangle-free and this paper is an investigation into various properties and interrelationships among the parameters of such designs.
Since y divides k, let us write k = my, m an integer. In [2], Baartmans and Shrikhande showed that 2~ m ~<y + 1. They also characterized the extremal cases. The case m = 2 is a Hadamard 3-design, while m =y + 1 is associated with the extension problem of symmetric designs (see [4]) studied by Cameron. Following [7], we call D exceptional if 3 <~ m ~ y.
The main part of this paper is an attempt to prove the following conjecture: There are only finitely many exceptional triangle-free quasi-symmetric designs. A computer search carded out in [2], supports this conjecture. We also show that the parameters of any triangle-free quasi-symmetric design can be expressed in terms of just two parameters y and m.
Section 2 gives some preliminary results required throughout the paper. The