Separating Pairs of Points by Standard Boxes

Noga Alon, Z. Füredi, M. Katchalski

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

Let A be a set of distinct points in ℝd. A 2-subset {a, b} of A is called separated if there exists a closed box with sides parallel to the axes, containing a and b but no other points of A. Let s(A) denote the number of separated 2-sets of A and put f(n, d) = max {s(A): A ⊂ ℝd, |A| = n}. We show that f(n, 2) = [n2/4] + n − 2 for all n≥2 and that for each fixed dimension d f(n,d)=(1−1/2 2 d−1−1)⋅n2/2+o(n2).

Original language English 205-210 6 European Journal of Combinatorics 6 3 https://doi.org/10.1016/S0195-6698(85)80028-7 Published - Jan 1 1985

Denote
Distinct
Closed
Subset
Standards

ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics

Cite this

Separating Pairs of Points by Standard Boxes. / Alon, Noga; Füredi, Z.; Katchalski, M.

In: European Journal of Combinatorics, Vol. 6, No. 3, 01.01.1985, p. 205-210.

Research output: Contribution to journalArticle

Alon, Noga ; Füredi, Z. ; Katchalski, M. / Separating Pairs of Points by Standard Boxes. In: European Journal of Combinatorics. 1985 ; Vol. 6, No. 3. pp. 205-210.
@article{09b89e7bbc1e4971ba7d9665cabcee0b,
title = "Separating Pairs of Points by Standard Boxes",
abstract = "Let A be a set of distinct points in ℝd. A 2-subset {a, b} of A is called separated if there exists a closed box with sides parallel to the axes, containing a and b but no other points of A. Let s(A) denote the number of separated 2-sets of A and put f(n, d) = max {s(A): A ⊂ ℝd, |A| = n}. We show that f(n, 2) = [n2/4] + n − 2 for all n≥2 and that for each fixed dimension d f(n,d)=(1−1/2 2 d−1−1)⋅n2/2+o(n2).",
author = "Noga Alon and Z. F{\"u}redi and M. Katchalski",
year = "1985",
month = "1",
day = "1",
doi = "10.1016/S0195-6698(85)80028-7",
language = "English",
volume = "6",
pages = "205--210",
journal = "European Journal of Combinatorics",
issn = "0195-6698",
publisher = "Academic Press Inc.",
number = "3",

}

TY - JOUR

T1 - Separating Pairs of Points by Standard Boxes

AU - Alon, Noga

AU - Füredi, Z.

AU - Katchalski, M.

PY - 1985/1/1

Y1 - 1985/1/1

N2 - Let A be a set of distinct points in ℝd. A 2-subset {a, b} of A is called separated if there exists a closed box with sides parallel to the axes, containing a and b but no other points of A. Let s(A) denote the number of separated 2-sets of A and put f(n, d) = max {s(A): A ⊂ ℝd, |A| = n}. We show that f(n, 2) = [n2/4] + n − 2 for all n≥2 and that for each fixed dimension d f(n,d)=(1−1/2 2 d−1−1)⋅n2/2+o(n2).

AB - Let A be a set of distinct points in ℝd. A 2-subset {a, b} of A is called separated if there exists a closed box with sides parallel to the axes, containing a and b but no other points of A. Let s(A) denote the number of separated 2-sets of A and put f(n, d) = max {s(A): A ⊂ ℝd, |A| = n}. We show that f(n, 2) = [n2/4] + n − 2 for all n≥2 and that for each fixed dimension d f(n,d)=(1−1/2 2 d−1−1)⋅n2/2+o(n2).

UR - http://www.scopus.com/inward/record.url?scp=85014877613&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014877613&partnerID=8YFLogxK

U2 - 10.1016/S0195-6698(85)80028-7

DO - 10.1016/S0195-6698(85)80028-7

M3 - Article

VL - 6

SP - 205

EP - 210

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 3

ER -