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Statement

Setting (Section 1, pp. 167-168). Points in the plane are in general position when no three lie on a line and no four on a circle (p. 167). For points $x_1,\ldots,x_n$ in general position, $d(x_i)$ is the number of distinct distances from $x_i$ to the other points, and $D(n)=\max_i d(x_i)$. Erdős notes that trivially $d(x_i)\ge(n-1)/3$ for every $i$ (p. 168).

Conjecture (3) (p. 168). There is an absolute constant $c>0$, independent of $n$ and of the position of the points, such that every set of $n$ points in general position has $D(n)>(1+c)n/3$. Erdős states it as his conviction ("I am sure that there is an absolute constant $c>0$") and gives no proof.

Question (4) (p. 168). Is there a set $x_1,\ldots,x_n$ in general position with $D(n)<(1-c)n$?

Weaker hypotheses (p. 168). Erdős writes that he "got nowhere with (3) and (4)", and suggests that (3) perhaps remains true when one assumes only that no four of the points are on a circle, or even only that no circle centred at one of the $x_i$ passes through more than three of the other $x_i$. He also asks (display (5)) to prove or disprove $\sum_{i=1}^n d(x_i)>(1+c)n^2/3$.

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 1, displays (3)--(5), printed p. 168, with the definition of general position on p. 167.

Read depth. Claims checked: the definitions, displays (3)--(5) and the remark on weaker hypotheses were read clause by clause on the page images of pp. 167-168.

Proof pointer

None; (3) and (4) are posed as open problems. The trivial bound $d(x_i)\ge(n-1)/3$ is stated without argument; it follows from general position, since no circle about $x_i$ meets more than three of the other points.

Dependencies

None.

Bears on

• Problem 654: the site's $f(n)$ is the least value of $\max_i d(x_i)$ over $n$-point sets with no four points on a circle, the first of the weaker hypotheses Erdős suggests for (3). Its question $f(n)>(1/3+c)n$ is (3) under that hypothesis, and its question $f(n)>(1-o(1))n$ asks that (4) have a negative answer for every $c>0$ and large $n$ under that hypothesis. The paper's (3) and (4) assume general position, which also excludes three points on a line, so a configuration with three points on a line bears on the site's questions but not on (3) or (4) as printed.