Unique distancing: oscarcunningham.com/670/unique

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.

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

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