描述集合论专门研究「无限集合」的可测性与阶层：可测集合在上层，完全不可测（病态）集合在下层。其历史可追溯到康托 1874 年证明「无限有不同大小」。作者用勒贝格测度对比：区间 (0,1) 的长度为 1、(0,10) 的长度为 10，虽同为无限且基数相同，却在「长度」上不同。

核心例子是圆周上的无限图：每次沿圆周前进固定距离；若距离是分数（如 1/5），走 5 步回到起点形成闭环；若不是分数，单个连通分支就无限延伸，且整体由无限多个彼此不连通的分支组成。若用两色交替著色，看似可行，但需对每个分支先「选」一个起点，等同使用选择公理（文中称其为 9 个基础公理之一），导致得到的颜色点集不可测。改以连续方式沿弧段上色，最后会剩一小段无法用两色收尾，必须引入第 3 色，才能得到可测的颜色集合。

Bernshteyn 在 2019 年一场分散式演算法演讲中注意到：有限网路的「局部演算法」也会出现「2 色很低效、允许 3 色则可高效」的门槛现象，与集合论中可测著色所需颜色数的门槛相呼应。他在 2023 年建立桥梁：把某类无限集合／无限图问题改写为电脑网路通讯与局部演算法问题，并证明即使图是「不可数」无限、无法全域唯一编号，也总能在任意有限标签数下做到「邻域内不重复」的重用标签，从而把高效局部演算法转成勒贝格可测的著色；反向翻译也带来新的难度估计与对问题分类的整理。

Descriptive set theory classifies infinite sets by how “measurable” they are: well-behaved sets sit high in the hierarchy, while pathological, nonmeasurable sets sit at the bottom. The article traces this viewpoint back to Cantor’s 1874 discovery of different infinities, and contrasts cardinality with Lebesgue measure: (0,1) has measure 1 and (0,10) has measure 10 even though both are infinite and share the same cardinality.

A key example builds an infinite graph on a circle by stepping a fixed arc-length each time. If the step is rational (e.g., 1/5), you return after 5 steps and get a loop; if it is irrational, each connected component is an infinite path, and there are infinitely many disconnected components. A naive 2-coloring implicitly uses the axiom of choice (one of nine foundational axioms) to pick a starting node in every component, producing color classes that are not Lebesgue-measurable. A continuous “arc-coloring” avoids choice but forces a 3rd color to finish the leftover segment, yielding measurable color sets.

Bernshteyn noticed in 2019 that distributed “local algorithms” on finite networks show thresholds where 2 colors can be extremely inefficient while 3 colors allow efficient solutions, mirroring measurability thresholds for infinite-graph colorings. In 2023 he proved a translation principle: efficient local algorithms correspond to Lebesgue-measurable colorings on the descriptive-set-theory side, despite uncountably many nodes. The key is reusable labeling: for any finite label set and any fixed neighborhood size, labels can always be assigned so nearby nodes differ. The bridge now supports new results, complexity estimates, and a more systematic classification of problems across both fields.