Abstract :
A Kirkman square with index λ, latinicity μ, block size k and v points, KSk(v; μ, λ), is a t × t array (t = λ(v − 1/μ(k − 1)) defined on a v-set V such that (1) every point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k, λ)-BIBD. The existence of KS2(v; μ, λ) has been completely settled. The existence of a first class for k = 3, KS3(v; 1, 2), was recently established in Lamken (1995). In this paper, we determine the spectrum of KS3(v; 2, 4) with at most two possible exceptions for v. Our main construction uses partitioned generalized balanced tournament designs; it is the first case of a very general construction for KSk(v; μ, λ).
