陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「On sets defining few ordinary lines」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Ben Green and I have just uploaded to the arXiv our new paper “ On sets defining few ordinary lines “, submitted to Discrete and Computational Geometry . This paper asymptotically solves two old questions concerning finite configurations of points in the plane . Given a set of points in the plane, define an ordinary line to be a line containing exactly two points of . The classical Sylvester-Gallai theorem , first posed as a problem by Sylvester in 1893, asserts that as long

It is then natural to pose the question of what is the minimal number of ordinary lines that a set of non-collinear points can generate. In 1940, Melchior gave an elegant proof of the Sylvester-Gallai theorem based on projective duality and Euler’s formula , showing that at least three ordinary lines must be created; in 1951, Motzkin showed that there must be ordinary lines. Previously to this paper, the best lower bound was by Csima and Sawyer , who in 1993 showed that there

已知结果和反例

As first observed by Böröczky, variants of this example also give few ordinary lines for odd , though not quite as few as ; more precisely, when one can find a configuration with ordinary lines, and when one can find a configuration with ordinary lines. Our first main result is that these configurations are best possible for sufficiently large :

Theorem 1 (Dirac-Motzkin conjecture) If is sufficiently large, then any set of non-collinear points in the plane will define at least ordinary lines. Furthermore, if is odd, at least ordinary lines must be created.

证明或构造的主线

The Dirac-Motzkin conjecture asserts that the first part of this theorem in fact holds for all , not just for sufficiently large ; in principle, our theorem reduces that conjecture to a finite verification, although our bound for “sufficiently large” is far too poor to actually make this feasible (it is of double exponential type). (There are two known configurations for which one has ordinary lines, one with (discovered by Kelly and Moser ), and one with (discovered by Crowe

Our second main result concerns not the ordinary lines, but rather the -rich lines of an -point set – a line that meets exactly three points of that set. A simple double counting argument (counting pairs of distinct points in the set in two different ways) shows that there are at most

阅读时建议盯住的点

-rich lines. On the other hand, on an elliptic curve , three distinct points P,Q,R on that curve are collinear precisely when they sum to zero with respect to the group law on that curve. Thus (as observed first by Sylvester in 1868), any finite subgroup of an elliptic curve (of which one can produce numerous examples, as elliptic curves in have the group structure of either or ) can provide examples of -point sets with a large number of -rich lines ( , to be precise). One ca

This problem was known as the Orchard planting problem , and was given a more poetic formulation as such by Jackson in 1821 (nearly fifty years prior to Sylvester!):

值得单独记下的条目

• lies on the union of an irreducible cubic curve and an additional points.
• lies on the union of an irreducible conic section and an additional lines, with of the points on in either of the two components.
• lies on the union of lines and an additional points.
• lies on the union of an irreducible cubic curve and an additional points.
• lies on the union of an irreducible conic section, a line, and an additional points, with of the points on in either of the first two components.
• lies on the union of a single line and an additional points.
• is a finite subgroup of an elliptic curve (EDIT: as pointed out in comments, one also needs to allow for finite subgroups of acnodal singular cubic curves), possibly shifted by a third root of unity.
• is the Borozcky example mentioned previously (the union of equally spaced points on the circle, and points on the line at infinity).

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源