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Problem 827 — Erdős Problems

Problem 827

Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\binom{k}{3}$ triples determine circles of different radii.

Determine $n_k$ .

Source: erdosproblems.com/827

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Er75h: Erdős, P., Some problems on elementary geometry. Austral. Math. Soc. Gaz. (1975), 2-3.
Er78c: Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. (1978), 52-54.
Er92e: Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
MaRo15: Martínez, L. and Roldán-Pensado, E., Points defining triangles with distinct circumradii. Acta Math. Hungar. 145 (2015), no. 1, 136-141.

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Status: Open
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