Cone dependence - A basic combinatorial concept

Ahlswede R, Khachatrian LH (2003)
In: Designs, Codes and Cryptography. Designs, Codes and Cryptography, 29(1-3). KLUWER ACADEMIC PUBL: 29-40.

Konferenzbeitrag | Veröffentlicht | Englisch
 
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Autor*in
Ahlswede, RudolfUniBi; Khachatrian, Levon H.
Abstract / Bemerkung
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.
Stichworte
combinatorial extremal problems; positive linear; combinations; Turan problem; binary sequences
Erscheinungsjahr
2003
Titel des Konferenzbandes
Designs, Codes and Cryptography
Band
29
Ausgabe
1-3
Seite(n)
29-40
ISSN
0925-1022
Page URI
https://pub.uni-bielefeld.de/record/1611224

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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.
Ahlswede, R., & Khachatrian, L. H. (2003). Cone dependence - A basic combinatorial concept. Designs, Codes and Cryptography, Designs, Codes and Cryptography, 29, 29-40. KLUWER ACADEMIC PUBL. doi:10.1023/A:1024183804420
Ahlswede, R., and Khachatrian, L. H. (2003). “Cone dependence - A basic combinatorial concept” in Designs, Codes and Cryptography Designs, Codes and Cryptography, vol. 29, (KLUWER ACADEMIC PUBL), 29-40.
Ahlswede, R., & Khachatrian, L.H., 2003. Cone dependence - A basic combinatorial concept. In Designs, Codes and Cryptography. Designs, Codes and Cryptography. no.29 KLUWER ACADEMIC PUBL, pp. 29-40.
R. Ahlswede and L.H. Khachatrian, “Cone dependence - A basic combinatorial concept”, Designs, Codes and Cryptography, Designs, Codes and Cryptography, vol. 29, KLUWER ACADEMIC PUBL, 2003, pp.29-40.
Ahlswede, R., Khachatrian, L.H.: Cone dependence - A basic combinatorial concept. Designs, Codes and Cryptography. Designs, Codes and Cryptography. 29, p. 29-40. KLUWER ACADEMIC PUBL (2003).
Ahlswede, Rudolf, and Khachatrian, Levon H. “Cone dependence - A basic combinatorial concept”. Designs, Codes and Cryptography. KLUWER ACADEMIC PUBL, 2003.Vol. 29. Designs, Codes and Cryptography. 29-40.