A remark on the problem of nonnegative k-subset sums
Problems of Information Transmission
Bibliography of publications by Rudolf Ahlswede
Information Theory, Combinatorics, and Search Theory
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We call A ⊂ \Bbb{E}n cone independent of B ⊂ \Bbb{E}n, 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 ⊂ \Bbb{E}n : A is cone independent} and their maximal cardinalities c(n) ≜ max{|A| : A ∈ P(n)}.We show that limn → ∞ \frac{c(n)}{2^n} ½, but can't decide whether the limit equals 1.Furthermore, for integers 1 k n we prove first results about cn (k, ℓ) ≜ max{|A| : A ∈ Pn(k, ℓ)}, where Pn (k, ℓ) = {A : A ⊂ Vnk and Vnℓ is cone independent of A} and Vnk equals the set of binary sequences of length n and Hamming weight k. Finding cn (k, ℓ) is in general a very hard problem with relations to finding Turan numbers.