Designs with no three mutually disjoint blocks

Let D(v,b,r,k,λ) be any quasi-symmetric block design with block intersection numbers 0 and y. Suppose D has no three mutually disjoint blocks. We show that for a given value of y, there are only finitely many parameter sets of such designs. Moreover, the ‘extremal’ designs D have one of the following parameter sets: (1) v = 4y, k = 2y, λ = 2y ? 1 (y 2) (2) v = y(y²+3y+1), k = y(y+1), λ =y²+y?1(y2) (3) v = (y+1)(y²+2y?1), k = y(y+1), λ =y² (y2) A computer search revealed only three parameter sets in the range 1y199, which are not of the above types.