At most 10 cylinders mutually touch: a Ramsey-theoretic approach

Abstract

Littlewood asked for the maximum number $N$ of congruent infinite cylinders that can be arranged in ${\mathbb{R}}^3$ so that every pair touches. We improve upon the proof of the second author that $N\le 18$ to show that $N\le 10$. Together with the lower bound established by Bozóki, Lee, and Rónyai, this shows that $N\in \{7,8,9,10\}$. Our method is based on linear algebra and Ramsey theory, and makes partial use of computer verification. We also provide a completely computer-free proof that $N\le 12$.

10 pages, comments welcome