  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.
