# Cone dependence - A basic combinatorial concept

Ahlswede R, Khachatrian L (2003)
In: DESIGNS CODES AND CRYPTOGRAPHY. 29(1-3). KLUWER ACADEMIC PUBL: 29-40.

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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.
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Ahlswede R, Khachatrian L. Cone dependence - A basic combinatorial concept. In: DESIGNS CODES AND CRYPTOGRAPHY. Vol 29. KLUWER ACADEMIC PUBL; 2003: 29-40.
Ahlswede, R., & Khachatrian, L. (2003). Cone dependence - A basic combinatorial concept. DESIGNS CODES AND CRYPTOGRAPHY, 29(1-3), 29-40. doi:10.1023/A:1024183804420
Ahlswede, R., and Khachatrian, L. (2003). “Cone dependence - A basic combinatorial concept” in DESIGNS CODES AND CRYPTOGRAPHY, vol. 29, (KLUWER ACADEMIC PUBL), 29-40.
Ahlswede, R., & Khachatrian, L., 2003. Cone dependence - A basic combinatorial concept. In DESIGNS CODES AND CRYPTOGRAPHY. no.29 KLUWER ACADEMIC PUBL, pp. 29-40.
R. Ahlswede and L. Khachatrian, “Cone dependence - A basic combinatorial concept”, DESIGNS CODES AND CRYPTOGRAPHY, vol. 29, KLUWER ACADEMIC PUBL, 2003, pp.29-40.
Ahlswede, R., Khachatrian, L.: Cone dependence - A basic combinatorial concept. DESIGNS CODES AND CRYPTOGRAPHY. 29, p. 29-40. KLUWER ACADEMIC PUBL (2003).
Ahlswede, Rudolf, and Khachatrian, L. “Cone dependence - A basic combinatorial concept”. DESIGNS CODES AND CRYPTOGRAPHY. KLUWER ACADEMIC PUBL, 2003.Vol. 29. 29-40.
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