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.
References
[1]
Croft, H.T., Falconer, K.J. and Guy, R.K. (2012) Unsolved Problems in Geometry. Springer Science & Business Media, Berlin.
[2]
ErdÖs, P. (1986) On Some Metric and Combinatorial Geometric Problems. Discrete Mathematics, 60, 147-153.
http://dx.doi.org/10.1016/0012-365X(86)90009-9
[3]
Palasti, I. (1989) Lattice-Point Examples for a Question of ErdÖs. Periodica Mathematica Hungarica, 20, 231-235.
http://dx.doi.org/10.1007/BF01848126
[4]
Palasti, I. (1989) A Distance Problem of P. ErdÖs with Some future Restrictions, North-Holland. Discrete Mathematics, 76, 155-156.
http://dx.doi.org/10.1016/0012-365X(89)90309-9
[5]
Liu, A. (1987) On the “Seven Points Problem” of P. ErdÖs. Studia Scientiarum Mathematicarum Hungarica, 22, 447-448.
[6]
Beck, M. and Geoghegan, R. (2010) The Art of Proof: Basic Training for Deeper Mathematics. Springer, New York.
[7]
Lee, G.Y. and Asci, M. (2012) Some Properties of the (p,q)-Fibonacci and (p,q)-Lucas Polynomials. Journal of Applied Mathematics, 2012, Article ID: 264842.
