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Claim. In Points defining triangles with distinct circumradii, a set of points in the plane is in general position when no four lie on a line or on a circle, and $n_k$ is the least integer such that any $n_k$ such points contain $k$ points all of whose triples determine circles of distinct radii. Theorem 1.1 proves that $n_k$ exists for every $k$ and that $n_k=O(k^9)$. Theorem 1.2 proves $n_4\le9$ and $n_5\le37$. The proof of Theorem 1.1 follows the scheme of Erdős's 1978 argument and handles the case that argument misses with Bézout's theorem on the intersections of two algebraic curves. Section 2 shows where Erdős's argument, which claimed $n_k\le k+2\binom{k-1}{2}\binom{k-1}{3}$, fails.

Covers. The existence of $n_k$ for every $k$, which is the question Erdős asked in 1975, the polynomial bound $n_k=O(k^9)$, and the bounds $n_4\le9$ and $n_5\le37$. The authors' convention, no four points on a line or a circle, is weaker than the one of Problem 827, no three points on a line and no four on a circle. Every set in general position under the problem's convention is in general position under the paper's, so the bounds hold for the problem's $n_k$ too. The paper does not determine $n_k$ for any $k$.

Acceptance. The paper appeared in Acta Math. Hungar. 145 (2015), no. 1, 136–141, which is the refereed evidence. The site's commentary credits the corrected argument and the $k^9$ bound to the paper, but the site labels the problem OPEN, so that credit is not acceptance. The library card is Martínez and Roldán-Pensado 2015.