Maximal and minimal coverings of (k− 1)-tuples byk-tuples

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Abstract

For m ^ k, an (m, k) system is a set of A>tuples (A -subsets) of 1,2, , m. A minimal (m, k) system is an (m, k) system with the property that every (k -l)-tuple of the m elements appears in at least one &-tuple of the system, but no system with fewer ^-tuples has this property. The numbers of fe-tuples in a minimal (m, k) system will be denoted by N k (m). A maximal (m, k) is an (m, k) system with the property that no (k -l)-tuple appears in more than one &-tuple of the system, but no system with more A -tuples has this property. The number of A -tuples in a maximal (m, k) system is D k (m). In this paper we shall be concerned with evaluating N k and D k and investigating the properties of extremal (m, k) systems for k -2, 3, and 4.

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