避免从有限个起点出发的四项等差数列
Avoiding four-term progressions from finitely many starts
- Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对任意有限起点集合，构造整数的一个双射枚举，使其在出现顺序中避免首项属于该集合的四项等差数列，并推广到前n项同时避免，通过有限分支前提关系和显式有界势函数实现，证明了有限前提闭包性并给出扩展定理。
AI中文摘要：
对于每个有限集合 $A\subset\mathbb{Z}$，我们构造一个从 $\mathbb{N}_0$ 到 $\mathbb{Z}$ 的双射 $p$，它以 $0,1$ 开头，并且在出现顺序中不包含首项属于 $A$ 的四项等差数列。更一般地，对于每个 $n\ge2$，该枚举可以从 $0,1,3,\ldots,2^{n-1}-1$ 开始，并同时避免所有起始于 $A$ 或这些前 $n$ 项中的此类等差数列。该构造使用有限分支的前提关系和一个显式的有界整数势函数。这证明了有限前提闭包性，并给出了一个穷举枚举，而不仅仅是一个全序。我们还给出了一个关于具有兼容二进制尾部约束的有限前缀的扩展定理。这些结果并未确定整数的每个枚举是否都包含有序的四项等差数列。
英文摘要：
For every finite set $A\subset\mathbb{Z}$, we construct a bijection $p:\mathbb{N}_0\to\mathbb{Z}$ beginning with $0,1$ that contains no four-term arithmetic progression, in occurrence order, whose first value belongs to $A$. More generally, for each $n\ge2$ the enumeration can begin with $0,1,3,\ldots,2^{n-1}-1$ and simultaneously avoid every such progression starting in $A$ or among these first $n$ entries. The construction uses finitely branching prerequisite relations and an explicit bounded integer potential. This proves finite prerequisite closure and gives an exhaustive enumeration, rather than merely a total order. We also give an extension theorem for finite prefixes with compatible binary-tail constraints. These results do not determine whether every enumeration of the integers contains an ordered four-term arithmetic progression.
