Widths and Heights of (0,1) -Matrices

Abstract

A number of combinatorial problems may be regarded as particular instances of the following rather general situation. Given a set X composed of n elements x1, x2, ..., xn, and m subsets X1 X2, … , Xm of X, find a minimal system of representatives for X1, X2, … , Xm. That is, single out a subset X* of X such that Xi ∩ X* is non-empty for i = 1,2, … ,m, and no subset of X containing fewer elements than X* has this property. To illustrate, each of the following can be thought of in these terms.