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「Bounds for the first few density Hales-Jewett numbers, and related quantities」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 thread is a continuation of the previous thread here on the polymath1 project . Currently, activity is focusing on the following problems:

Let be the largest size of a set in without a combinatorial line . Thanks to efforts from previous threads, we have the first five values of :

已知结果和反例

We also know that , and are working to narrow this further. The arguments justifying these bounds can be found here . The latest bounds for the next few values of can be found here .

Consider a variant of the above problem in which each element of with a 1s, b 2s, and c 3s is weighted by the factor ; this gives a total weight of . Let be the largest weight of a line-free set of , and let be the largest size of a subset of

证明或构造的主线

which contains no upward-pointing equilateral triangles with . It is known that ; the “ hyper-optimistic conjecture ” is that one in fact has . This would imply density Hales-Jewett for k=3.

Currently, we know the following values for these sequences:

阅读时建议盯住的点

There are also some further upper and lower bounds known; see this spreadsheet for the latest bounds, and this page for proofs of the bounds. The data so far is consistent with the conjecture, but more work would be needed to obtain a more convincing case for it.

Moser’s cube problem asks to compute the largest size of a subset of the cube without geometric lines. The first few values of are known:

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