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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.'

joriki
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Ayush
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  • 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. – Alex B. Apr 19 '11 at 00:34
  • How what is possible? – Qiaochu Yuan Apr 19 '11 at 00:35
  • 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. – joriki Apr 19 '11 at 00:37
  • I haven't got down to paper and pen with this, but would it not be possible to place points in a circle with integer distances between any two points? – Ayush Apr 19 '11 at 00:39
  • @xbonez: no. If you put more than $2 \pi r$ points on a circle of radius $r$, there will be two points at a distance less than $1$ from each other by the pigeonhole principle. – Qiaochu Yuan Apr 19 '11 at 00:42
  • But since we're given an infinite number of points, how can we restrict the circle to have a radius of r? Couldn't I say that if you place more than n points on a straight line of length 'n' (where n is an integer), the distance between two consecutive points is not an integer? – Ayush Apr 19 '11 at 00:44
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    You can't have a circle of infinite radius in euclidean space. You can though have an infinite line. – JSchlather Apr 19 '11 at 00:46
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    @xbonez: what reasonable interpretation of "circle of infinite radius" does not evaluate to "line"? – Qiaochu Yuan Apr 19 '11 at 01:01
  • It is possible for a (fixed) circle to have infinitely many points, each at a rational distance from all the others (but of course not at an integral distance). – Gerry Myerson Apr 19 '11 at 01:03

2 Answers2

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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.

user642796
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Gerry Myerson
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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.)

user642796
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Omar Antolín-Camarena
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    I believe this is the proof given in the Yaglom and Yaglom book "Challenging Problems..." –  Apr 19 '11 at 06:18