Christian Lawson-Perfect @christianp

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Unique distancing: https://oscarcunningham.com/670/unique-distancing-problem/

How many points can you place in an $n \times n$ grid so that all pairwise distances are distinct? The linked post concerns whether $n$ points are possible (no for all but finitely many cases because there are too many pairs and too few sums of squares) but it also looks interesting to maximize the number of points.

Oscar Cunningham's Blog Unique Distancing Problem | Oscar Cunningham's Blog By Oscar Cunningham

Jul 16, 2021, 01:01 AM··Web

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**domotorp** @domotorp · Jul 16, 2021

Jul 16, 2021

domotorp @domotorp

@11011110 See the problem here too: https://math.stackexchange.com/questions/1208087/maximum-number-of-points-you-can-put-on-grid-n-times-m-with-no-equidistant

math.stackexchange.comMaximum number of points you can put on grid $ n\times m$ with no equidistant?Assume we have a grid of $n\times m$ points. and the distance between two rows or two columns is 1 ( unit ). I have a couple of questions related to this grid:- What is the list of possible length

**0xDE** @11011110 · Jul 16, 2021

Jul 16, 2021

0xDE @11011110

@domotorp Thanks — a comment there led to the relevant paper https://arxiv.org/abs/1211.1776 — its bounds are far from tight but I didn't see any improvements in later work citing it.

arxiv.orgA note on distinct distance subsetsIt is shown that given a set of $N$ points in the plane or on the sphere, there is a subset of size $\gtrsim N^{1/3}/\log N$ with all pairwise distances between points distinct.

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