A Kochen-Specker System has at least 22 vertices

• Bas Westerbaan (bas@westerbaan.name)
• Sander Uijlen (suijlen1337@gmail.com)

Abstract

At the heart of the Conway-Kochen Free Will theorem and Kochen and Specker's argument against non-contextual hidden variable theories is the existence of a Kochen-Specker (KS) system: a set of points on the sphere that has no {0, 1}-coloring such that at most one of two orthogonal points are colored 1 and of three pairwise orthogonal points exactly one is colored 1. In public lectures, Conway encouraged the search for small KS systems. At the time of writing, the smallest known KS system has 31 vectors. Arends, Ouaknine and Wampler have shown that a KS system has at least 18 vectors, by reducing the problem to the existence of graphs with a topological embeddability and non-colorability property. The bottleneck in their search proved to be the sheer number of graphs on more than 17 vertices and deciding embeddability. Continuing their effort, we prove a restriction on the class of graphs we need to consider and develop a more practical decision procedure for embeddability to improve the lower bound to 22. In 2022, Li, Bright, and Ganesh improved the lower bound to 23, then shortly after to 24 in 2023 using a SAT solver + Computer Algebra System approach to dramatically reduce the search space.

This repository

This repository contains the

• LaTeX sourcecode of the preprint (under /paper);
• LaTeX sourcecode for the slides used at QPL2014 (under /qpl2014);
• Data generated by some of the computations (under /graphs);
• source code used for the computation (under /code, in particular /code/comp<n>.py) and
• source code of the website kochen-specker.info (under /site).
  • First, in /code, run python generateSite.py.
  • Then, in /site, run jekyll build.
  • Find the site in /site/_site.