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)) 2n subsets of [n] and we give tight bounds on the o (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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics