Sperner理论是组合数学的一个分支,其研究对象是偏序集,主要考虑偏序集上满足一些限定条件的极值问题,它起源于Sperner在1928年提出的一个定理：在子集格中,最大秩集构成基数最大的反链。Sperner 引理的提出引起了数学家们广泛的关注,在经过接近一个世纪的发展后,Sperner理论已经发展成为一门系统完善的理论。本文首先介绍了Sperner引理的内容及其提出背景,然后通过两道数学竞赛题目引出Sperner引理的两种证明方法,并对它们做出了简评。在阐述了偏序集和对称链的内容后,我们梳理了对称链的证法。此外,文章还介绍了布尔矩阵理论的证法和图论的证法。

Sperner theory is a branch of combinatorics , which studies partially ordered sets , mainly considering extreme value problems on partially ordered sets that meet some qualifying conditions , which originated from a theorem proposed by Sperner in 1928 : in a subset lattice , the largest rank set forms the largest cardinal backlink. The introduction of Sperner's lemma has attracted widespread attention among mathematicians, and after nearly a century of development, Sperner's theory has developed into a systematic and well-developed theory. This paper first introduces the content of Sperner's lemma and its background, and then introduces the two proof methods of Sperner's lemma through two mathematical competition questions, and briefly comments on them. After elaborating on the contents of partial order sets and symmetric chains, we sort out the dialectic of symmetric chains. In addition, the article introduces the proof of Boolean matrix theory and the proof of graph theory.