Erdős 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 ra…

Paul Erdős posed the problem and later claimed a positive answer. The question asks for the smallest number $n_k$ guaranteeing $k$ points whose triples define circles with pairwise distinct radii.

Known results

• Erdős (1978) claimed $n_k\le k+\binom{k-1}{2}\binom{k-1}{3}$, but the proof omitted a nontrivial case.
• Erdős (1985) restated the claim and gave partial credit to E. Straus.
• The earlier argument also yielded the related bound $n_k\le 2\binom{k-1}{2}\binom{k-1}{3}+k$.
• Explicit small cases recorded in the literature include $n_4\le 9$ and $n_5\le 37$.

Bézout-based repair

The note “Points defining triangles with distinct circumradii” claims to repair Erdős’s gap using Bézout’s theorem and proves the polynomial upper bound $n_k=O(k^9)$. This establishes substantial progress but does not determine the exact value of $n_k$; the claim is not independently verified here.

Current status (as of August 2026): A polynomial upper bound and small-case bounds are claimed, while the exact value of $n_k$ remains open.