A (v, k, t) covering design, or covering, is a family of k-subsets, called blocks, chosen from a v-set, such that each t-subset is contained in at least one of the blocks. The number of blocks is the covering's size, and the minimum size of such a covering is denoted by C(v, k, t). It is easy to see that a covering must contain at least (vt)/(kt) blocks, and in 1985 Rödl  proved a long-standing conjecture of Erdos and Hanani  that for fixed k and t, coverings of size (vt)/(kt)(1 + 0(1)) exist (as v → ∞). An earlier paper by the first three authors  gave new methods for constructing good coverings, and gave tables of upper bounds on C(v, k, t) for small v, k, and t. The present paper shows that two of those constructions are asymptotically optimal: For fixed k and t, the size of the coverings constructed matches Rödl's bound. The paper also makes the 0(1) error bound explicit, and gives some evidence for a much stronger bound.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics