On the number of pairwise touching cylinders in $\mathbb{R}^d$

TL;DR

This paper investigates the maximum number of mutually touching infinite cylinders of unit radius in higher-dimensional Euclidean spaces, providing exponential upper bounds and a polynomial-based approach.

Contribution

It extends the problem of touching cylinders from 3D to higher dimensions and introduces a flexible polynomial method to derive exponential upper bounds.

Findings

Established exponential upper bounds on the number of touching cylinders in $\

Developed a polynomial-based method to analyze cylinder contact configurations in higher dimensions.

Abstract

John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.''} Boz\'oki, Lee, and R\'onyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in $\mathbb{R}^3$ can mutually touch. We consider this problem in higher dimensions, and obtain exponential (in $d$) upper bounds on the number of mutually touching cylinders in $\mathbb{R}^d$. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and non-equalities.