TITLE:
On the ErdÖs Distance Conjecture in Geometry
AUTHORS:
Amir Jafari, Amin Najafi Amin
KEYWORDS:
ErdÖs Distance Problem, General Position, Graph Pattern, Distance Distribution
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.6 No.3,
June
21,
2016
ABSTRACT: ErdÖs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.