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「A massively collaborative mathematical project」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

My good friend Tim Gowers has just started an experimental “ massively collaborative mathematical project ” over at his blog . The project is entitled “ A combinatorial approach to density Hales-Jewett “, and the aim is to see if progress can be made on this problem by many small contributions by a large number of people, as opposed to the traditional model of a few very large contributions by a small number of people (see this article for more on the “rules of the game”, and

I can describe the problem here. Let n be a large integer, and let be the set of all strings of length n using the alphabet , thus for instance . A combinatorial line in is a triple of points in that can be formed by taking a string of length n using the alphabet with at least one occurrence of the “wildcard” x, and then substituting the values of 1, 2, 3 for the wildcard. For instance, the string would lead to the combinatorial line in . The (k=3) case of the density Hales-J

已知结果和反例

Density Hales-Jewett theorem. Let . Then if n is sufficiently large depending on , every subset of of density at least contains a combinatorial line.

[Furstenberg and Katznelson handled the case of general k in a subsequent paper . The k=1 case is trivial, and as pointed out in this post by Gil Kalai , the k=2 case follows from Sperner’s theorem .]

证明或构造的主线

Furstenberg and Katznelson’s proof uses ergodic theory, and in particular does not obviously give any bound as to how large n has to be depending on before the theorem takes effect. No other proofs of this theorem are currently known. So it would be desirable to have a combinatorial proof of the k=3 density Hales-Jewett theorem. Since this theorem implies Roth’s theorem , and Roth’s theorem has a combinatorial proof based on the triangle removal lemma (see e.g. my Simons lect

Further articles on this project are collected at this page .

阅读时建议盯住的点

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

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