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Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid question and point me in the right direction?

'Given an infinite set of points in a plane, if the distance between any two points is an integer, prove that all these points lie on a straight line.'

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  • $\begingroup$ Can we turn your question around? What is (or might be) invalid about the above problem? I am having trouble imagining where your doubts might be coming from. $\endgroup$
    – Alex B.
    Apr 19, 2011 at 0:34
  • $\begingroup$ How what is possible? $\endgroup$ Apr 19, 2011 at 0:35
  • $\begingroup$ Please use more descriptive titles. The title is meant for people to be able to see what questions are about from looking at the titles. $\endgroup$
    – joriki
    Apr 19, 2011 at 0:37
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    $\begingroup$ You can't have a circle of infinite radius in euclidean space. You can though have an infinite line. $\endgroup$
    – JSchlather
    Apr 19, 2011 at 0:46
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    $\begingroup$ @xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"? $\endgroup$ Apr 19, 2011 at 1:01

2 Answers 2

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MR0013511 (7,164a) Anning, Norman H.; Erdős, Paul Integral distances. Bull. Amer. Math. Soc. 51, (1945). 598–600.

The authors show that for any n there exist noncollinear points $P_1,\dots,P_n$ in the plane such that all distances $P_iP_j$ are integers; but there does not exist an infinite set of non-collinear points with this property. Reviewed by I. Kaplansky

I can add that the first result mentioned requires lots of points to be on a circle. I believe the current record for points in the plane with all distances integers, no three on a line, no 4 on a circle, is 8.

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  • $\begingroup$ The article and a follow-up. $\endgroup$ Apr 19, 2011 at 1:26
  • $\begingroup$ @J.M. It appears both links point the same place, probably the follow-up $\endgroup$ Apr 19, 2011 at 1:44
  • $\begingroup$ The article: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/… $\endgroup$
    – JavaMan
    Apr 19, 2011 at 1:49
  • $\begingroup$ I'm no longer confident that an 8-point configuration is known, but a 7-point configuration is given by Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021). $\endgroup$ Apr 19, 2011 at 1:51
  • $\begingroup$ @Ross: yes, I copied wrongly; DJC gives the correct link. $\endgroup$ Apr 19, 2011 at 1:55
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Say $A$, $B$ and $C$ are three points with pairwise integer distances. Any other point $P$ with integer distances from those three will satisfy $|AP-PB| \leq AB$ and $|AP-PC| \leq AC$, by the triangle inequality. So $P$ lies on an intersection of a hyperbola $|AP-BP| = k$ (for some integer $k \leq AB$) and a hyperbola $|AP-PC| = m$ (for some integer $m \leq AC$). There are finitely many such points $P$ (at most $4\;AB\;AC$).

(I'd be surprised if the argument in the Anning and Erdős paper is very different from this one.)

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    $\begingroup$ I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..." $\endgroup$
    – user641
    Apr 19, 2011 at 6:18

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