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Statement

Setting (p. 2). $f(n)$ is the largest number of distinct circles of radius one determined by the $\binom n3$ triples of $n$ distinct points in the plane, read as the maximum over all such sets of points; see display (1) for the definition and the bounds $3n/2<f(n)\le n(n-1)$.

Display (2) (p. 2). Erdős writes that he is sure that

$$\frac{f(n)}{n^2}\to0\quad\text{and}\quad\frac{f(n)}{n}\to\infty,\qquad(2)$$

the limits taken as $n\to\infty$. He states that he was not able to prove (2). He adds that he expects an asymptotic formula for $f(n)$ to be very difficult and that there may be no simple exact expression for $f(n)$.

The paper then poses variants (pp. 2-3): determine or estimate $f(n)$ for points in general position, meaning no four on a circle and no three on a line; estimate or determine $g(n)$, the maximum number of triples with circumscribed circle of unit radius when not all the points lie on one unit circle, with the guess that the maximum occurs when $n-1$ of the points lie on a unit circle; determine $g(n)$ when the points are in general position; and estimate or determine $h(n)$, defined for $n$ given points in the plane in general position as the largest integer such that at least $h(n)$ circles of different radii pass through three of the points, and say how $h(n)$ changes when it is only assumed that not all the points lie on a circle.

Proof pointer

None: the paper states (2) as a conjecture and proves neither half.

Read depth. Claims checked: display (2), the sentence after it, and the variants on pp. 2-3 were read clause by clause on the print.

Source. P. Erdős, Some problems on elementary geometry, Austral. Math. Soc. Gaz. 2 (1975), 2--3, pp. 2-3. The edition read is identified on the source card.

Dependencies

The definition of $f(n)$ before display (1).

Bears on

• Problem 104: the first half of (2), $f(n)/n^2\to0$, with $f(n)$ read as the maximum over $n$-point sets, is the problem's statement that $n$ points in the plane determine $o(n^2)$ distinct unit circles containing at least three of them. The paper records it as a conjecture and proves nothing about it. The second half, $f(n)/n\to\infty$, is not part of the problem's statement.