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arXiv · 2510.03924

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

Travis Dillon·Junnosuke Koizumi·Sammy Luo
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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 $N \leq 18$ to show that $N \leq 10$. 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 $N \leq 12$.

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Travis Dillon, Junnosuke Koizumi, Sammy Luo. 2025-10-04. At most 10 cylinders mutually touch: a Ramsey-theoretic approach. https://arxiv.org/abs/2510.03924

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