Vol. 26, No. 1, 1968

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ISSN: 0030-8730
Maximal and minimal coverings of (k 1)-tuples by k-tuples

J. G. Kalbfleisch and Ralph Gordon Stanton

Vol. 26 (1968), No. 1, 131–140
Abstract

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

Mathematical Subject Classification
Primary: 05.04
Milestones
Received: 11 August 1967
Published: 1 July 1968
Authors
J. G. Kalbfleisch
Ralph Gordon Stanton