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): 900-999 (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 700-799 thread of the polymath1 project, which is now full. During the course of that thread, we have made significant progress on the three problems being focused on:
Let be the largest size of a set in without a combinatorial line . We now have both human and computer-assisted proofs of the first few values of this sequence:
已知结果和反例
The current best-known bounds for are . Given the gap involved here, and the rate at which the complexity of the problem has increased with n, it seems unlikely that 下面会 be able to compute exactly any time soon, but it is possible that some improvement can still be made 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 r>0. It is known that ; the “ hyper-optimistic conjecture ” is that one in fact has . This would imply density Hales-Jewett for k=3.
Currently, the conjecture is verified for , where the values of for are 1,2,4,6,9,12 respectively; see this page and this page for details. It seems feasible to handle . Currently we know that and .
阅读时建议盯住的点
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:
The best asymptotic lower bound known is still of the order of . Improving this bound seems related to the well-known problem of improving the bounds in Behrend’s construction of an AP-3 free set of integers.
阅读和落地时建议先做的 5 件事
- 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。
- 找一个最小反例或边界情形，确认假设少一条会怎样。
- 把证明拆成可独立检验的引理，每步只保留一个新想法。
- 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
- 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
常见问题 FAQ
什么是AI智能系统？
「AI智能系统」可概括为：This is a continuation of the 700-799 thread of the polymath1 project, which is now full. During the course of that thread, we have made significant progress on the three problems 本文从定义、方法与实践要点展开说明。
为什么要关注AI智能系统？
关注AI智能系统，是因为它直接影响效率、风险与可复制性。文中指出：This is a continuation of the 700-799 thread of the polymath1 project, which is now full. During the course of that thread, we have made significant progress on the three problems being focused on:
如何落地AI智能系统？有哪些关键步骤？
建议按以下路径推进AI智能系统：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。。细节见正文对应章节。
AI智能系统适合哪些人或团队？
AI智能系统更适合：产品/技术负责人、运营与增长团队、需要落地智能体或自动化的中小团队、关注「AI智能系统」方向的读者。若你只需要单次聊天式问答，可先读概念；若要上生产，请重点看步骤、权限与风控相关段落。
关于「问题在问什么」，本文给出了什么结论？
在「问题在问什么」部分，要点是：t, which is now full. During the course of that thread, we have made significant progress on the three problems being focused on: Let be the largest size of a set in without a combinatorial line . We now have both human
关于「已知结果和反例」，本文给出了什么结论？
在「已知结果和反例」部分，要点是：time soon, but it is possible that some improvement can still be made 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