Cone Dependence - A Basic Combinatorial Concept

We call A ⊂ En cone independent of B ⊂ En , the euclidean n-space, if no a = (a1, . . . , an) ∈ 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 ⊂ {0, 1}n ⊂ En : A is cone independent} and their maximal cardinalities c(n) max{|A| : A ∈ P(n)}. We show that limn→∞ c(n) 2n > 1 2 , but can’t decide whether the limit equals 1. Furthermore, for integers 1 < k < ≤ n we prove first results about cn(k, ) max{|A| : A ∈ Pn(k, )}, where Pn(k, ) = {A : A ⊂ V n k and V n is cone independent of A} and V n k equals the set of binary sequences of length n and Hamming weight k. Finding cn(k, ) is in general a very hard problem with relations to finding Turan numbers.