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 [Cone dependence-a basic combinatorial concept, Preprint 00-117, Diskrete Strukturen in der Mathematik SFB 343, Universitat Bielefeld, 2000] and the result has geometric consequences. (C) 2007 Elsevier B.V. All rights reserved.