On the existence of one-sided designs

A collection of subsets (called blocks) of a fixed vertex set (possibly with repetition) is called a (t _n , t _n –1, ..., t ₁; a _m , a _m –1, ..., a ₁)-design if it satisfies certain regularity conditions on the number of blocks which contain subsets of the vertex set of certain size, and other regularity conditions on the size of the intersections of certain numbers of the blocks. (For example, a BIBD (or (b, v, r, k, )-configuration) is a (1, 2; 1)-design, and a t-design is a (t, t–1, ..., 1; 1)-design.) A design has design-type (t _n , ..., t ₁; a _m , ..., a ₁) if it satisfies only those conditions. A one-sided design is a design with design-type (t _n , ..., t ₁;) or (;a _m , ..., a ₁). In this paper we show, by construction, that any one-sided design-type is possible.