The structure of triangle-free quasi-symmetric designs

Abstract

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 pⁿ, then p = 2 and D is a Hadamard design.