# On Sets of Distances of n Points

```@article{Erds1946OnSO,
title={On Sets of Distances of n Points},
author={Paul Erd{\"o}s},
journal={American Mathematical Monthly},
year={1946},
volume={53},
pages={248-250}
}```
• P. Erdös
• Published 1946
• Mathematics
• American Mathematical Monthly
1. The function f(n). Let [P. ] be the class of all planar subsets P. of n points and denote by f(n) the minimum number of different distances determined by its n points for P,, an element of { P. }. Clearly, f(3) = 1 (with the three points forming the vertices of an equilateral triangle) f(4) = 2, f(5) = 2. The following theorem establishes rough bounds for arbitrary n. Though I have sought to improve this result for many years, I have not been able to do so.
510 Citations
The number of distinct distances from a vertex of a convex polygon
• Mathematics, Computer Science
• J. Comput. Geom.
• 2013
The present note slightly improves on the best known lower bound due to Dumitrescu (2006), an improved bound on the maximum number of isosceles triangles determined by P. Expand
On the Number of Maximal Regular Simplices Determined by n Points in Rd
• Mathematics
• 2003
A set V = {x 1,…, x n } of n distinct points in Euclidean d-space ℝ d determines 2 n distances ∥x j − x i ∥ (1 ≤ i < j ≤ n). Some of these distances may be equal. Many questions concerning theExpand
The grid revisted
• Computer Science, Mathematics
• Discret. Math.
• 1993
It is proved that if P is a set of n points in a disk of radius n such that the minimum distance between them is 1, and |P|/n?∞, then the set of angles determined by these points is everywhere dense in 0,2?]. Expand
The Number of Different Distances Determined by n Points in the Plane
• F. Graham
• Computer Science
• J. Comb. Theory, Ser. A
• 1984
It is shown that f(n) > cn " ' for some fixed constant c and this has remained for 30 years as the best lower bound known for f( n). Expand
The number of different distances determined by n points in the plane
Abstract A classical problem in combinatorial geometry is that of determining the minimum number f(n) of different distances determined by n points in the Euclidean plane. In 1952, L. Moser provedExpand
The number of different distances determined by a set of points in the Euclidean plane
• Computer Science, Mathematics
• Discret. Comput. Geom.
• 1992
It is shown that the inequality of the minimum numberd(n) of different distances determined by a set ofn points in the Euclidean plane is best possible as is shown by the lattice points inThe plane. Expand
Large Subsets of Points with all Pairs ( Triples ) Having Different Distances ( Areas )
Let {p1, . . . , pn} ⊆ R. We think of d ≤ n. How big is the largest subset X of points such that all of the distances determined by elements of ( X 2 ) are different? We show that X is at leastExpand
On the diameter of separated point sets with many nearly equal distances
A point set is separated if the minimum distance between its elements is 1. We call two real numbers nearly equal if they differ by at most 1. We prove that for any dimension d ≥ 2a nd any γ> 0, if PExpand
On the diameter of separated point sets with many nearly equal distances
• Computer Science, Mathematics
• Eur. J. Comb.
• 2006
It is proved that for any dimension d ≥ 2 and any γ > 0, if P is a separated set of n points in Rd such that at least γn2 pairs in (P 2) determine nearly equal distances, then the diameter of P is at least C(d, γ)n2/(d-1) for some constant C( d,γ) > 0. Expand
Repeated Angles in the Plane and Related Problems
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
• 1992
Abstract We show that a set of n points in the plane determine O(n2 log n) triples that define the same angle α, and that for many angles α (including π 2 ) this bound is tight in the worst case. WeExpand

#### References

SHOWING 1-3 OF 3 REFERENCES
On Sets of Distances of N Points in Euclidean Space
Let [Pg)] he the clans of all subsets Pjlc, of the Ic dimensional space consisting of TZ distinct points and having diameter 1. Denote by g,Jn, T) the maximum number of times a given distance r canExpand
On sets of distances of 7t points, this MONTHLY
• 1946
On a problem of Sidon, jour
• London Math. Sot.,
• 1941