Cone Dependence—A Basic Combinatorial Concept

Abstract

We call A ⊂ $\mathbb{E}$ ⁿ cone independent of B ⊂ $\mathbb{E}$ ⁿ, the euclidean n-space, if no a = (a ₁,..., a _n ) ∈ 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 ⊂ $\mathbb{E}$ ⁿ : A is cone independent} and their maximal cardinalities c(n) ≜ max{|A| : A ∈ P(n)}.

We show that lim _{n → ∞} $\frac{{c\left( n \right)}}{{2^n }}$ > $\frac{1}{2}$, 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) ≜ max{|A| : A ∈ P _n (k, ℓ)}, where P _n (k, ℓ) = {A : A ⊂ V ⁿ _k and V ⁿ _ℓ is cone independent of A} and V ⁿ _k 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.