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Statement

Setting (p. 52). Points in the plane are in general position when no three lie on a line and no four on a circle. The paper recalls its 1975 question: for every $k$, is there an $n_k$ such that among any $n_k$ points in general position one can always find $k$ of them all of whose $\binom{k}{3}$ triples determine circles of different radii.

Inequality 1 (p. 52). The paper asserts that a simple argument gives

$$n_k\le k+2\binom{k-1}{2}\binom{k-1}{3},$$

which would make $n_k$ exist for every $k$. It adds that (1) is probably very far from best possible.

The argument as printed (p. 52), in outline. Take $m=k+2\binom{k-1}{2}\binom{k-1}{3}$ points in general position and a maximal subset $x_1,\ldots,x_\ell$ all of whose triples determine circles of different radii, and suppose $\ell<k$. The paper asserts that maximality gives, for each remaining point $x_u$, a circle through $x_u$ and two points $x_i,x_j$ of the subset whose radius is one of the $\binom{\ell}{3}$ radii already occurring among the subset's triples. At most two circles of a given radius pass through two given points, so the remaining $m-\ell$ points lie on at most $2\binom{\ell}{2}\binom{\ell}{3}$ circles, and general position puts at most one of them on each such circle. Hence $m\le\ell+2\binom{\ell}{2}\binom{\ell}{3}\le k-1+2\binom{k-1}{2}\binom{k-1}{3}$, a contradiction. The print writes $n-\ell$ for the number of remaining points, where $m-\ell$ is meant.

The gap. Adding $x_u$ to a maximal subset can also fail because two new triples through $x_u$ determine circles of the same radius, a coincidence the argument does not treat. Martínez and Roldán-Pensado identify this case in Section 2 of their note, as recorded on the source card of their paper, and repair the argument with Bézout's theorem, proving $n_k=O(k^9)$ under a weaker general-position condition. The bound (1) itself is therefore unproved by this paper.

Source. P. Erdős, Some more problems on elementary geometry, Austral. Math. Soc. Gaz. 5 (1978), no. 2, 52--54: the question, inequality (1) and its argument on p. 52. The edition read is identified on the source card.

Read depth. Claims checked: the definition, the inequality and the argument were read clause by clause on the page image of p. 52. The gap is reported from the Martínez and Roldán-Pensado source card, not from a reading of their paper here.

Proof pointer

Page 52, the paragraph after (1), as outlined above; the argument is incomplete for the reason given under The gap.

Dependencies

None beyond elementary facts: at most two circles of a given radius pass through two given points.

Bears on

• Problem 827: the problem asks for the value of $n_k$ under the same general-position condition. The paper claims the upper bound (1), which would show that $n_k$ exists; the argument is incomplete, and the problem page records no claim for it. Existence and a polynomial bound come from the later Martínez and Roldán-Pensado paper.