Witness Sets for Families of Binary Vectors

Given a family ~ ~ {0, 1 } "' of binary vectors of length m, a set W ~ { I ..... m} is called a witness set for r ~ .~, if for all other r' • ~. there exists a coordinate c • W such that r,.~r',.. The smallest cardinality of a witness set for r •b~ is denoted w(r) = t ry( r ) . In this note we show that Y~r~ w(r)= O(1,~'13/2) and constructions are given to show that this bound is tight. Further information is derived on the distribution of values of { w(r) I r e.~}. ~ 1996 Academic Press. Inc.