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Source. The paragraphs spanning pp. 52--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

Definition (p. 52). For $n$ distinct points in the plane, $f_2(n)$ is the largest number of pairs at distance $1$.

Reported bounds (p. 52).

1. Erdős (1946): $f_2(n)<2n^{3/2}$.
2. The lattice points in the plane give $f_2(n)>n^{1+c/\log\log n}$, from the number of representations of an integer as a sum of two squares. The print does not specify the constant $c$.
3. Szemerédi proved $f_2(n)/n^{3/2}\to0$, for which Erdős had offered a prize.
4. Beck and Spencer proved $f_2(n)<n^{3/2-c}$, again with an unspecified constant $c$.

Conjecture and prize (pp. 52--53). Erdős thinks the lower bound is the right one. He says that $f_2(n)<n^{1+\varepsilon}$ "is nowhere in sight" and offers a prize for it.

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 90: the lecture records the lattice lower bound $n^{1+c/\log\log n}$, Erdős's belief that it is the right one, which is the problem's question, and the upper bounds known in 1982. It proves nothing toward the problem.