## Separating pairs of points of standard boxes.(English)Zbl 0592.05002

Authors’ summary: ”Let A be a set of distinct points in $${\mathbb{R}}^ d$$. 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 $$f(n,d)=\max \{s(A): A\subset {\mathbb{R}}^ d$$, $$| A| =n\}$$. We show that $$f(n,2)=[n^ 2/4]+n-2$$ for all $$n\geq 2$$ and that for each fixed dimension d $f(n,d) = (1-1/2^{2^{d-1}-1})\cdot n^ 2/2+o(n^ 2).''$
Reviewer: J.Carter

### MSC:

 05A05 Permutations, words, matrices 05C99 Graph theory

### Keywords:

distinct points; separated 2-sets
Full Text:

### References:

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