Lower Bounds for Transversal Covers

Authors:
Brett Stevens;Lucia Moura;Eric Mendelsohn
Affiliations:
Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3;Department of Computer Science, University of Toronto, Toronto, Canada M5S 3G4;Department of Mathematics, University of Toronto, Toronto, Canada M5S 3G3
Venue:
Designs, Codes and Cryptography
Year:
1998

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Abstract

A transversal cover is a set of gkpoints in k disjoint groups of size gand a collection of b transversal subsets, calledblocks, such that any pair of points not contained in the samegroup appears in at least one block. A central question is todetermine, for given g, the minimum possible bfor fixed k, or, alternatively, the maximum kfor fixed b. The case g=2 was investigatedand 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 understoodfor small values of k. Constructions exist butthese only produce upper bounds on b. The presentarticle is concerned with lower bounds on b. Wedevelop three general lower bounds on b for fixedg and k. The first one is proved usingone of the principal constructions brett:97a, the second comesfrom the study of intersecting set-systems, and the third isshown by a set packing argument. In addition, we investigateupper bounds on k for small fixed b.This proves useful to reduce or eliminate the gap between lowerand upper bounds on b for some transversal coverswith small k.