A Kochen-Specker (KS) set is a finite collection of vectors on the two-sphere containing no antipodal pairs for which it is impossible to assign 0s and 1s such that no two orthogonal vectors are assigned 1 and exactly one vector in every triplet of mutually orthogonal vectors is assigned 1. The existence of KS sets lies at the heart of Kochen and Specker's argument against non-contextual hidden variable theories and the Conway-Kochen free will theorem. Identifying small KS sets can simplify these arguments and may contribute to the understanding of the role played by contextuality in quantum protocols. In this paper we derive a weak lower bound of 10 vectors for the size of any KS set by studying the opposite notion of large non-KS sets and using a probability argument that is independent of the graph structure of KS sets. We also point out an interesting connection with a generalisation of the moving sofa problem around a right-angled hallway on the two-sphere.

翻译：Kochen-Specker（KS）集是二维球面上不含对径点对的有限向量集合，其不可能为每个向量分配0或1，使得无两个正交向量同时被赋值为1，且每个由相互正交向量构成的三元组中恰有一个向量被赋值为1。KS集的存在性构成了Kochen与Specker反对非语境隐变量理论的核心论据，也是Conway-Kochen自由意志定理的基础。辨识小型KS集可简化上述论证，并可能有助于理解语境性在量子协议中的作用。本文通过研究大型非KS集的对立概念，并采用独立于KS集图结构的概率论证方法，推导出任意KS集大小的弱下界为10个向量。此外，我们指出了该问题与二维球面上直角走廊中移动沙发问题泛化形式之间的有趣联系。