AI 中文总结
该研究确定了平面上非共线非共圆n点集确定的含三点不同圆的最小数量c(n)，除三个特殊阶数外等于F(n)，还解决了无三点共线的变体问题，相关工作由人类与AI合作完成。
AI 中文摘要
对于n≥4，设c(n)为欧氏平面上既非共线也非共圆的n点集中，包含至少三个点的不同圆的最小数量。令F(n)=1+组合数(n-1,2)-向下取整((n-1)/2)。我们确定了所有n≥4时的c(n)：除三个特殊阶数外，它等于F(n)。我们还解决了无三点共线的变体问题，该变体仅有一个特殊阶数。证明和精确有限验证由人类研究者与人工智能系统合作完成。
英文摘要
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.
Comments27 pages, 5 figures; ancillary files contain exact verification code and Lean 4 formalizations
