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Source. The paragraphs spanning pp. 41--43 of P. Erdős, Combinatorial problems in geometry, Math. Chronicle 12 (1983), 35--54, the transcript of an invited address at the 17th New Zealand Mathematics Colloquium (Dunedin, 17--19 May 1982), as named on the source card. The lecture numbers none of its statements; the pages are the journal's own.

Statement

Theorem (pp. 41--42, joint with Anning). If an infinite set of points in the plane has every pairwise distance an integer, then all of its points lie on one straight line.

Read depth. Claims checked: the passage was read clause by clause on the page images of the print. A second reader checked the statement, hypotheses, label and page against the print.

Proof pointer

The lecture presents the short proof Erdős says he found later at Kaplansky's prodding (pp. 42--43). In outline: if the set is not collinear it contains a non-degenerate triangle $ABC$; for any further point $X$ of the set the integer $XB-XC$ lies between $-a$ and $a$, where $a=BC$, so $X$ lies on one of at most $2a+1$ hyperbolas with foci $B,C$, and likewise on one of at most $2b+1$ hyperbolas with foci $A,C$, where $b=AC$. Two such hyperbolas meet in at most four points, so there are at most $4(2a+1)(2b+1)$ such points $X$, a bound depending only on the triangle, which contradicts infinitude.

Dependencies

None.

Bears on

• Problem 213: context. The theorem rules out infinite integral-distance sets off a line; the problem asks about finite sets in general position, which the lecture poses next (problem_p43).