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Statement

Question (p. 3, unnumbered). Erdős asks (quoted) "Is it true that to every $k$ there is an $n_k$ so that if there are given $n_k$ points in the plane in general position one can always find $k$ of them so that all the $\binom k3$ triples determine circles of different radii?" He adds that at present he cannot even prove that $n_k$ exists.

The question itself does not define general position. The paper's earlier definition for a set of points (p. 2) is no four on a circle and no three on a line; for three points alone (p. 2) it is not on a line.

Context in the paper (p. 3). Erdős says he was led to the question by E. Klein's (Mrs. E. Szekeres's) problem whether for every $k$ there is an $m_k$ such that any $m_k$ points in the plane with no three on a line contain the vertices of a convex $k$-gon. He recalls that she proved $m_4=5$, that Turán and Makai proved $m_5=9$, and that G. Szekeres conjectured $m_k=2^{k-2}+1$.

Proof pointer

None: the paper poses the question and proves nothing about it.

Read depth. Claims checked: the question, the remark after it and the paragraph on Klein's problem were read clause by clause on p. 3 of the print.

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

Dependencies

None.

Bears on

• Problem 827: the question is the existence of the number $n_k$ whose least value the problem asks to determine, under the same condition on the $k$ chosen points. The paper proves nothing about it, and Erdős says he cannot prove that $n_k$ exists.