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「The story of Erdős problem #1026」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

Problem 1026 on the Erdős problem web site recently got solved through an interesting combination of existing literature, online collaboration, and AI tools. The purpose of this blog post is to try to tell the story of this collaboration, and also to supply a complete proof.

The original problem of Erdős, posed in 1975 , is rather ambiguous. Erdős starts by recalling his famous theorem with Szekeres that says that given a sequence of distinct real numbers, one can find a subsequence of length which is either increasing or decreasing; and that one cannot improve the to , by considering for instance a sequence of blocks of length , with the numbers in each block decreasing, but the blocks themselves increasing. He also noted a result of Hanani that

已知结果和反例

This problem was added to the Erdős problem site on September 12, 2025, with a note that the problem was rather ambiguous. For any fixed , this is an explicit piecewise linear function of the variables that could be computed by a simple brute force algorithm, but Erdős was presumably seeking optimal bounds for this quantity under some natural constraint on the . The day the problem was posted, Desmond Weisenberg proposed studying the quantity , defined as the largest constant

Though not stated on the web site, one can formulate this problem in game theoretic terms. Suppose that Alice has a stack of coins for some large . She divides the coins into piles of consisting of coins each, so that . She then passes the piles to Bob, who is allowed to select a monotone subsequence of the piles (in the weak sense) and keep all the coins in those piles. What is the largest fraction of the coins that Bob can guarantee to keep, regardless of how Alice divides

证明或构造的主线

AI-generated images continue to be problematic for a number of reasons, but here is one such image that somewhat manages at least to convey the idea of the game:

For small , one can work out by hand. For , clearly : Alice has to put all the coins into one pile, which Bob simply takes. Similarly : regardless of how Alice divides the coins into two piles, the piles will either be increasing or decreasing, so in either case Bob can take both. The first interesting case is . Bob can again always take the two largest piles, guaranteeing himself of the coins. On the other hand, if Alice almost divides the coins evenly, for instance into pil

阅读时建议盯住的点

An hour after Desmond’s comment, Stijn Cambie noted (though not in the language I used above) that a similar construction to the one above, in which Alice divides the coins into pairs that are almost even, in such a way that the longest monotone sequence is of length , gives the upper bound . It is also easy to see that is a non-increasing function of , so this gives a general bound . Less than an hour after that, Wouter van Doorn noted that the Hanani result mentioned above

The next day, Stijn computed the first few values of exactly:

值得单独记下的条目

• The sequence can be numerically computed as a sequence of rational numbers.
• When appropriately normalized and arranged, visible patterns in this sequence appear that allow one to conjecture the form of the sequence.
• This problem is a weighted version of the Erdős-Szekeres theorem.
• Among the many proofs of the Erdős-Szekeres theorem is the proof of Seidenberg in 1959, which can be interpreted as a discrete rectangle packing argument.
• This problem can be reinterpreted as a continuous square packing problem, and in fact is closely related to (a generalized axis-parallel form of) Erdős problem 106, which concerns such packings.
• The axis-parallel form of Erdős problem 106 was recently solved by Baek-Koizumi-Ueoro.
• The paper of Praton shows that Erdős Problem 106 implies the generalized version needed for this problem. This implication specializes to the axis-parallel case.

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