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Statement

Setting (p. 1). For $n$ distinct points $x_1,\ldots,x_n$ in $k$-dimensional Euclidean space $E_k$, $D_k(x_1,\ldots,x_n)$ is the number of distinct distances among them, and $f_k(n)$ is its minimum over all such sets of $n$ points. $P_k(n)$ is the largest integer such that some $n$ points of $E_k$ have $P_k(n)$ pairs $x_i,x_j$ with $d(x_i,x_j)=1$.

Conjecture (p. 2, display (4)). Erdős writes that he still believes his old conjectures

$$f_2(n)>cn/(\log n)^{1/2},\qquad P_2(n)<n^{1+c/\log\log n} \qquad(4)$$

to be true. The print gives no quantifier for the constant; the small letter in the exponent of the second inequality is printed as an italic letter that the scan does not clearly resolve between $c$ and $\varepsilon$, and either reading denotes an unspecified constant. He offers (p. 2, quoted) "$500 for a proof or disproof", and a separate prize for the weaker bound $P_2(n)<n^{1+\varepsilon}$.

Context in the paper

The conjectures follow the bounds the paper records (pp. 1-2): as the best results until recently, $P_2(n)=o(n^{3/2})$ (Szemerédi) and $f_2(n)>cn^{2/3}$ (L. Moser), displayed as (1); and as recent improvements, $P_2(n)<n^{3/2-c}$ for some $c>0$ (J. Beck and J. Spencer), displayed as (2), and $f_2(n)>cn^{5/7}$ (Fan Chung), displayed as (3). After (4) the paper also suggests (p. 2) that perhaps there is always a point $x_1$ with more than $cn/(\log n)^{1/2}$ distinct distances to the other points, and counts this among several conjectures discussed in its reference [1]. The paper proves none of these statements.

Read depth. Claims checked: the setting, display (4) and the prize sentence were read clause by clause on pp. 1-2 of the print.

Source. P. Erdős, Problems and results in combinatorial geometry, in Discrete geometry and convexity (New York, 1982), Ann. New York Acad. Sci. 440 (1985), 1-11, Section I, pp. 1-2. The edition read is identified on the source card.

Bears on

• Problem 89: the first inequality of (4), read as holding for some constant $c>0$ and all large $n$, is the affirmative answer to the problem's question whether every $n$ points in the plane determine $\gg n/\sqrt{\log n}$ distinct distances. The paper records it as a conjecture and proves nothing about it.
• Problem 90: the second inequality of (4), read as holding for some constant $c$ and all large $n$, is the affirmative answer to the problem's question whether every $n$ points in the plane have at most $n^{1+O(1/\log\log n)}$ pairs at distance one. The quoted offer is made for a proof or disproof of "my old conjectures" of (4), without saying whether it is one prize or one for each. The paper records the inequality as a conjecture and proves nothing about it.