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Statement

Anning-Erdős theorem (Section 6, p. 172). If $S$ is an infinite set in the plane such that the distance between any two of its points is an integer, then $S$ lies on a line. The paper cites W. H. Anning and P. Erdős, Integral distances, Bull. Amer. Math. Soc. 51 (1945), 598--600, and Erdős's note of the same title, ibid. 996.

Ulam's and Besicovitch's conjectures (p. 172). If the distances of the infinite set $S$ are only known to be rational, $S$ need not lie on a line, but Erdős expects it to have a very special structure. "Ulam conjectured 40 years ago that $S$ cannot be everywhere dense and Besicovitch (independently) conjectured that the set of limit points of $S$ cannot contain some convex $n$-gon for $n>n_0$."

The same section continues (pp. 172-173) with the question whether, for every $n$, there are $n$ points in general position (no three on a line, no four on a circle) with all $\binom n2$ distances integers; Erdős reports Lagrange's six such points (Fig. 2) and Harborth's message that he and Kemnitz showed it is the example of least diameter, and the only one of diameter at most $220$.

Source. P. Erdős, Some combinatorial and metric problems in geometry, Intuitive geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai 48, North-Holland, Amsterdam-New York, 1987, 167--177 (MR 89i:52012); Section 6, printed pp. 172-173, the conjectures on p. 172.

Read depth. Claims checked: the theorem as restated, the two conjectures and the general-position question were read clause by clause on the page images of pp. 172-173. The theorem's proof is in the cited 1945 papers and was not read here.

Proof pointer

None in this paper; the Anning-Erdős theorem is cited, and the two conjectures are posed without argument.

Dependencies

Anning and Erdős, Bull. Amer. Math. Soc. 51 (1945), 598--600, for the integral-distance theorem.

Bears on

• Problem 212: the site asks whether some dense subset of $\mathbb R^2$ has all pairwise distances rational; Ulam's conjecture as reported here is that no infinite plane set with all distances rational is everywhere dense, the negative answer.