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 ) ) 2 n subsets of [n] [ n ] and we give tight bounds on the o(1)2n o ( 1 ) 2 n term. The problem was posed by Ahlswede and Khachatrian [Cone dependencea basic combinatorial concept, Preprint 00-117, Diskrete Strukturen in der Mathematik SFB 343, Universitt Bielefeld, 2000] and the result has geometric consequences.
05d05 52b05