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For $n\in\mathbb N$, let $f(n)$ be the maximal number of times distance $1$ can occur among $n$ points in the plane: $$ f(n) = \max_{ \{ x_1,\ldots,x_n \} \subset \mathbb R^2} \# \big \{ i<j : \| x_i-x_j \| =1 \big \}. $$

What is the asymptotic growth of $f(n)$?
Is $\lim_{n\to \infty} f(n)/n=\infty$?

The function $f(n)$ starts very much like A047932:
0, 1, 3, 5, 7, 9, 12, 14, 16, 19, 21, 24, ... but eventually grows faster.

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  • $\begingroup$ @SamHopkins Indeeed, my question is a duplicate. What is the usual way of dealing with this? (I keep the question up, but it gets closed?) $\endgroup$
    – André Henriques
    Commented Jan 10, 2023 at 22:14
  • $\begingroup$ It can be closed as a duplicate if 5 people vote for that option for closure. $\endgroup$
    – Sam Hopkins
    Commented Jan 10, 2023 at 22:16
  • $\begingroup$ But you can also delete it, in fact, it think even the duplicate could be deleted as this is such a well-known problem that can be found with a minimal amount of searching online. Ps. Your function values are incorrect from 16. $\endgroup$
    – domotorp
    Commented Jan 10, 2023 at 22:18
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    $\begingroup$ @Sam: Entering the title of the question to google returns first the paper by Erdős where this question is formulated: users.renyi.hu/~p_erdos/1946-03.pdf Also at the top is this other mathoverflow question, asking the very same thing: mathoverflow.net/questions/355227/… $\endgroup$
    – domotorp
    Commented Jan 11, 2023 at 7:33
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    $\begingroup$ @domotorp I find the tone of your comment slightly hurtful. Yes - my question is duplicate. But it feels like you're almost angry at me... that you feel that, because I wasn't aware of Erdös' work, that makes me a bad person. Please be mindful of other members of the community (in this case me) when you post your comments. $\endgroup$
    – André Henriques
    Commented Jan 11, 2023 at 11:39

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