Large Convex Cones in Hypercubes

A family of subsets of [n] is positive linear combination free if the characteristic vector of neither member is the positive linear combination of the characteristic vectors of some other ones. We construct a positive linear combination free family which contains ( 1 − o ( 1 ) ) 2 n subsets of [n] and we give tight bounds on the o ( 1 ) 2 n term. The problem was posed by Ahlswede and Khachatrian [R. Ahlswede and L. Khachatrian, Cone dependence – a basic combinatorial concept, Preprint 00-117, Diskrete Strukturen in der Mathematik SFB 343, Universität Bielefeld, 2000] and the result has geometric consequences.