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Source. The first paragraph of p. 53 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

Problem and conjecture (p. 53). How many distinct distances do $n$ points in the plane determine? Erdős thinks the answer is $n/\sqrt{\log n}$, which the lattice points attain: by a theorem of Landau the number of distinct values $\sqrt{u^2+v^2}$ in the relevant range is of order $n/\sqrt{\log n}$ (the print writes "the number of distinct integers of the form $\sqrt{u^2+v^2}$ is $n/\sqrt{\log n}$"). He offers a prize for the problem.

Reported lower bounds (p. 53). Erdős originally proved that $n$ points determine $\sqrt n$ distinct distances; Leo Moser improved this to $n^{2/3}$, and Fan Chung very recently to $n^{5/7}$, with a proof Erdős calls very tricky. The print states the bounds without constants.

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 none of these; it reports them.

Dependencies

None.

Bears on

• Problem 89: Erdős's conjecture here is that the right order is $n/\sqrt{\log n}$, which includes the problem's lower bound $\gg n/\sqrt{\log n}$. The lecture records the lower bounds known in 1982 and proves nothing toward the problem.