# A note on distinct distance subsets

@article{Charalambides2013ANO, title={A note on distinct distance subsets}, author={Marcos Charalambides}, journal={Journal of Geometry}, year={2013}, volume={104}, pages={439-442} }

It is shown that given a set of N points in the plane, sphere or hyperbolic plane, there is a subset of size $${\gtrsim (N/\log N)^{1/3}}$$≳(N/logN)1/3 with all pairwise distances between points distinct.

#### 11 Citations

A note on distinct distances

- Computer Science, Mathematics
- Combinatorics, Probability and Computing
- 2020

Abstract We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of… Expand

Applications of the Canonical Ramsey Theorem to Geometry

- Mathematics
- 2013

Let P be a set of n points in R^d. How big is the largest subset X of P such that all of the distances determined between pairs are different? We show that X is at at least Omega(n^{1/6d}) This is… Expand

Chapter 9: Distinct distances variants

- 2015

Let subset(n) = min|P|=n subset(P). In other words, subset(n) is the maximum number satisfying the property that every set of n points in the plane contains a subset of subset(n) points that do not… Expand

1 FINITE POINT CONFIGURATIONS

- 2017

The study of combinatorial properties of finite point configurations is a vast area of research in geometry, whose origins go back at least to the ancient Greeks. Since it includes virtually all… Expand

Distinct Angle Problems and Variants

- Computer Science, Mathematics
- ArXiv
- 2021

It is shown that the number of distinct angles formed by n points in general position is O(n2), providing the first non-trivial bound for this quantity, and a new class of asymptotically optimal point configurations with no four cocircular points is introduced. Expand

Distinct Volume Subsets

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2015

The best known bound for the function h_{2,d}(n) is improved and it is shown that it is at least a power of $n$ for all positive integers a and d. Expand

Incidences between points and generalized spheres over finite fields and related problems

- Mathematics
- 2014

Let $\mathbb{F}_q$ be a finite field of $q$ elements where $q$ is a large odd prime power and $Q =a_1 x_1^{c_1}+...+a_dx_d^{c_d}\in \mathbb{F}_q[x_1,...,x_d]$, where $2\le c_i\le N$, $\gcd(c_i,q)=1$,… Expand

Distinct Distances: Open Problems and Current Bounds

- Mathematics, Computer Science
- ArXiv
- 2014

The variants of Erdýos’ distinct distances problem are surveyed and the current best bounds for each of those are surveyed. Expand

Extremal Problems in Analysis

- Mathematics
- 2014

In this thesis, we consider a selection of extremal problems which arise in mathematical analysis. The problems themselves come from a variety of fields of mathematics including the theory of… Expand

A survey of Elekes-R\'onyai-type problems

- Mathematics
- 2016

We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial f ∈ R[x, y] has a large image on a Cartesian product A×B ⊂… Expand

#### References

SHOWING 1-10 OF 10 REFERENCES

On Sets of Distances of n Points

- Mathematics
- 1946

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. }.… 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. We… Expand

Distinct Distances Determined By Subsets of a Point Set in Space

- Computer Science, Mathematics
- Comput. Geom.
- 1991

This work answers the question to determine the largest number fd(k) = f with the property that almost all k-element subsets of any n-element set in Rd determine at least f distinct distances, for all sufficiently large n. Expand

Isosceles Triangles Determined by a Planar Point Set

- Mathematics, Computer Science
- Graphs Comb.
- 2002

It is proved that, for any ɛ>0 and n>n0(ɛ), every set of n points in the plane has at most triples that induce isosceles triangles, which easily implies the best currently known lower bound, , for the smallest number of distinct distances determined by n points by Solymosi–Cs. Expand

Point sets with distinct distances

- Mathematics, Computer Science
- Comb.
- 1995

An efficient sequential algorithm for finding a subset of a given set with the desired properties, for example with distinct distances, of size as guaranteed by the probabilistic method under a more general setting is given. Expand

Extremal problems in discrete geometry

- Mathematics, Computer Science
- Comb.
- 1983

Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points. Expand

On distinct distances among points in general position and other related problems

- Mathematics, Computer Science
- Period. Math. Hung.
- 2008

It is shown that there exist n-element point sets in the plane in general position, and parallelogram free, that determine only O(n2/√log n) distinct distances, answering a question of Erdős, Hickerson and Pach. Expand

On the Erdos distinct distance problem in the plane

- Mathematics
- 2010

In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erd\"os. We follow the set-up of… Expand

Research problems in discrete geometry

- Computer Science
- 2005

This chapter discusses 100 Research Problems in Discrete Geometry from the Facsimile edition of the World Classics in Mathematics Series, vol. Expand