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OP-9203OpenGeometry and topology

Erdős #827 · geometry

Let nkn_k be minimal such that if nkn_k points in R2\mathbb{R}^2 are in general position then there exists a subset of kk points such that all (k3)\binom{k}{3} triples determine circles of different radii.

Determine $n_k…

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Problem statement

Let nkn_k be minimal such that if nkn_k points in R2\mathbb{R}^2 are in general position then there exists a subset of kk points such that all (k3)\binom{k}{3} triples determine circles of different radii.

Determine nkn_k.

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Background

In [Er75h] Erdős asks whether nkn_k exists. In [Er78c] he gave a simple argument which proves that it does, and in fact

nk≤k+2(k−12)(k−13),n_k \leq k+2\binom{k-1}{2}\binom{k-1}{3},

but this argument is incorrect, as explained by Martinez and Roldán-Pensado [MaRo15].

Martinez and Roldán-Pensado give a corrected argument that proves nk≪k9n_k\ll k^9. A simple probabilistic argument, given in the comments by FlaredRain, improves this to nk≪k5n_k\ll k^5.

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #827 link

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