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Source. The paragraphs spanning pp. 47--48 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

Question (pp. 47--48, Steinhaus, from Willy Moser's problem collection). Is there a set $S$ in the plane such that every set congruent to $S$ (every translate and rotation of it) contains exactly one lattice point, a point with integer coordinates?

Reformulation (p. 48). Such an $S$ contains no two points at distance $\sqrt{u^2+v^2}$ for integers $u,v$, since otherwise a congruent copy would contain two lattice points; conversely a set with no such distance has no congruent copy containing two lattice points. The question is then whether such a set can always contain one lattice point.

Erdős says he is almost certain that no such set exists, that he has no idea how to prove it, and that there may be a simple solution, though he does not think so.

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 paper proves nothing about the question beyond the reformulation.

Dependencies

None.

Bears on

• Problem 215: the question is the problem's question. The lecture records Erdős's expectation of a negative answer and proves nothing; the problem page records how the question was later resolved.