Cone dependence - A basic combinatorial concept

Ahlswede R, Khachatrian LH. Cone dependence - A basic combinatorial concept. In: Designs, Codes and Cryptography. Designs, Codes and Cryptography. Vol 29. KLUWER ACADEMIC PUBL; 2003: 29-40.We call A subset of E-n cone independent of B subset of E-n, the euclidean n-space, if no a = ( a(1),..., a(n)) is an element of A equals a linear combination of B \{a} with non-negative coefficients. If A is cone independent of A we call A a cone independent set. We begin the analysis of this concept for the sets P(n) = {A subset of {0, 1}(n) subset of E-n : A is cone independent} and their maximal cardinalities c(n) (=Delta) max{\A\ : A is an element of P( n)}. We show that lim(n-->infinity) c(n)/2n > 1/2, but can't decide whether the limit equals 1. Furthermore, for integers 1 < k < l = n we prove first results about c(n) ( k, l) =Delta max{\A\ : A is an element of P-n ( k, l)}, where P-n (k, l) = {A : A subset of V-k(n) and V-l(n) is cone independent of A} and V-k(n) equals the set of binary sequences of length n and Hamming weight k. Finding c(n) (k, l) is in general a very hard problem with relations to finding Turan numbers