Circles determined by planar point sets

Abstract

For $n\geq 4$, let $c(n)$ be the minimum number of distinct circles containing at least three points of an $n$-point set in the Euclidean plane, where the set is neither collinear nor concyclic. Put[F(n)=1+\binom{n-1}{2}-\left\lfloor\frac{n-1}{2}\right\rfloor.]We determine $c(n)$ for every $n\geq 4$: it equals $F(n)$ apart from three exceptional orders. We also solve the variant in which no three points are collinear; that variant has a single exceptional order. The proofs and exact finite verifications were developed through a collaboration between human researchers and artificial-intelligence systems.

Disclosure

“Eureka, an AI-assisted mathematical research system developed by the JiuChong team at the University of Science and Technology of China, supplied the initial numerical verification for n ≥ 16. After a human proof for n ≥ 92 had been found, OpenAI Codex helped sharpen it to the uniform range n ≥ 15 and carried out the exact small-order verifications and searches for exceptional configurations. The mathematical part of the argument has also been translated into Lean 4 and checked aga”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-22T03:09:28.605346Z

Structural counts