Discrepancy of Set-systems and Matrices

Abstract

The discrepancy of a set-system is the minimum number d for which the vertices can be 2-coloured red and blue so that in each. of the given sets, the difference between the numbers of red and blue vertices is at most d. In this paper. we introduce various mathematically more tractable variants of this notion. We prove several inequalities relating these numbers, and formulate several further conjectures. We extend the notion to a general matrix, and formulate it as a problem of covering the unit cube by convex bodies. © 1986, Academic Press Inc. (London) Limited. All rights reserved.