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「DHJ(3): 1100-1199 (Density Hales-Jewett type numbers)」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

This is a continuation of the 900-999 thread of the polymath1 project, which is now full. We’ve made quite a bit of progress so far on our original mission of bounding density Hales-Jewett numbers. In particular, we have shown

This timeline shows the history of these and other developments in the project.

已知结果和反例

There are still several numbers that look feasible to compute. For instance, the bounds on should be able to be narrowed further. Work is slowly progressing also on the equal-slices Hales-Jewett numbers , which are hyper-optimistically conjectured to equal the Fujimura numbers ; this has been verified by hand up to n=3 and by integer programming up to n=5. We are also looking at trying to reduce the dependence on computer assistance in establishing the result; the best human

Some progress has recently been made on some other related questions. For instance, we now have a precise description of the lower bound on coming from the Behrend-Elkin construction, namely

证明或构造的主线

Also, it is probably a good time to transport some of the discussion in earlier threads to the wiki and make it easier for outsiders to catch up. (Incidentally, we need a logo for that wiki; any suggestions would be welcome!)

Comments on this thread should be numbered starting at 1100.

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。

值得单独记下的条目

• : Any subset of with 125 points contains a geometric line ; and we have several examples with 124 points with no geometric line. (The proof is partly computer-assisted; details are here .)
• : A genetic algorithm has constructed 353-point solutions in with no geometric line; in the other direction, linear and integer programming methods have shown that any set with 362 points must have a geometric line.
• : In the triangular grid , any set of 41 points contains an upwards-pointing equilateral triangle with ; and we have 40-point examples without such triangles. This was conducted by an integer program.

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