Abstract
We say, that a subset K of the columns of a matrix is called a key, if every two rows that agree in the columns of K agree also in all columns of the matrix. A matrix represents a Sperner system K, if the system of minimal keys is exactly K. It is known, that such a representation always exists. In this paper we show, that the maximum of the minimum number of rows, that are needed to represent a Sperner system of only two element sets is 3(n/3+o(n)). We consider this problem for other classes of Sperner systems (e.g., for the class of trees, i.e. each minimal key has cardinality two, and the keys form a tree), too. The concept of keys plays an important role in database theory.
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Tichler, K. Extremal Theorems for Databases. Annals of Mathematics and Artificial Intelligence 40, 165–182 (2004). https://doi.org/10.1023/A:1026162114022
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DOI: https://doi.org/10.1023/A:1026162114022