Lower Bounds for Transversal Covers

A transversal cover is a set of gk points in k disjoint groups of size g and a collection of b transversal subsets, called blocks, such that any pair of points not contained in the same group appears in at least one block. A central question is to determine, for given g, the minimum possible b for fixed k, or, alternatively, the maximum k for fixed b. The case g=2 was investigated and completely solved by Sperner sperner:28, Rényi renyi:71, Katona katona:73, and Kleitman and Spencer kleitman:73. For arbitrary g, asymptotic results are known but little is understood for small values of k. Constructions exist but these only produce upper bounds on b. The present article is concerned with lower bounds on b. We develop three general lower bounds on b for fixedg and k. The first one is proved using one of the principal constructions brett:97a, the second comes from the study of intersecting set-systems, and the third is shown by a set packing argument. In addition, we investigate upper bounds on k for small fixed b. This proves useful to reduce or eliminate the gap between lower and upper bounds on b for some transversal covers with small k.