There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
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3$\begingroup$ See work related to Erdos' Unit Distance Problem: en.wikipedia.org/wiki/… $\endgroup$– Mark LewkoCommented Mar 19, 2020 at 2:40
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1$\begingroup$ $P(0)=P(1)=0; P(2)=1; P(3)=3; P(4)=5; P(5)=7; \ldots$ $\endgroup$– Wlod AACommented Mar 19, 2020 at 3:03
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$\begingroup$ $P(6)=9; P(7)=12; \ldots$ $\endgroup$– Wlod AACommented Mar 19, 2020 at 3:07
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$\begingroup$ I assume "unordered pairs of different points". $\endgroup$– Wlod AACommented Mar 19, 2020 at 3:22
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$\begingroup$ The subsection of the Wiki page @MarkLewko linked to changed its name to en.wikipedia.org/wiki/Unit_distance_graph#Number_of_edges $\endgroup$– Sam HopkinsCommented Jan 11, 2023 at 23:48
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The number is tabulated at OEIS. It seems that it's only known up to $n=14$ (and some scattered larger values). Links are given there to some papers on the topic. Evidently, no one knows how to do it for general $n$.
Also discussed on math.stackexchange.
answered Mar 19, 2020 at 5:05
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$\begingroup$ Hey, I am glad, thank you. I knew also P(8)=14 but didn't want to risk it. $\endgroup$– Wlod AACommented Mar 19, 2020 at 6:51
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2$\begingroup$ There aren't any values bigger than 14 for which it would be known, see the table at the end of www1.pub.informatik.uni-wuerzburg.de/eurocg2020/data/uploads/…. $\endgroup$– domotorpCommented Mar 19, 2020 at 8:58