Large convex cones in hypercubes

A family of subsets of [n][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))2n(1-o(1))2n subsets of [n][n] and we give tight bounds on the o(1)2no(1)2n 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, Universität Bielefeld, 2000] and the result has geometric consequences.