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

Travis Dillon, Junnosuke Koizumi, Sammy Luo

TL;DR

The paper tackles the champagne glass problem: determining the maximum $N$ of congruent infinite cylinders in $\mathbb{R}^3$ that pairwise touch, equivalently a set of lines at mutual distance $1$. It introduces a linear-algebraic realizability framework tied to a chirality function $\varepsilon(L,L')$ and shows that realizable graphs correspond to Gram-type matrices with at most $3$ negative eigenvalues, enabling non-realizability results for several graphs. By combining these obstructions with a Ramsey-theoretic statement on $K_{10}$ colorings (verified computationally) and a computer-free bound $N\le 12$, the authors prove $N\le 10$, placing the true value in $\{7,8,9,10\}$ when paired with the known lower bound $N\ge 7$. The work also outlines algorithmic approaches for the Ramsey step and provides a fully human-readable bound in a related setting, while discussing extensions to higher dimensions and related open questions.

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 $N \leq 12$.

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

TL;DR

The paper tackles the champagne glass problem: determining the maximum of congruent infinite cylinders in that pairwise touch, equivalently a set of lines at mutual distance . It introduces a linear-algebraic realizability framework tied to a chirality function and shows that realizable graphs correspond to Gram-type matrices with at most negative eigenvalues, enabling non-realizability results for several graphs. By combining these obstructions with a Ramsey-theoretic statement on colorings (verified computationally) and a computer-free bound , the authors prove , placing the true value in when paired with the known lower bound . The work also outlines algorithmic approaches for the Ramsey step and provides a fully human-readable bound in a related setting, while discussing extensions to higher dimensions and related open questions.

Abstract

Littlewood asked for the maximum number of congruent infinite cylinders that can be arranged in so that every pair touches. We improve upon the proof of the second author that to show that . Together with the lower bound established by Bozóki, Lee, and Rónyai, this shows that . 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 .

Paper Structure

This paper contains 5 sections, 12 theorems, 14 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem 2
  3. Concluding remarks
  4. Algorithm for Theorem 3
  5. Human-readable proof of Ch(3) <= 12

Key Result

Theorem 1

There is no configuration of 11 lines in $\mathbb{R}^3$ in which the distance between every pair of lines is exactly 1.

Figures (7)

  • Figure 1: $H_6$ (left) and $H_7$ (right)
  • Figure 2: $K_6\setminus C_5$ (left), $K_6\setminus H_6$ (middle), $K_7\setminus H_7$ (right)
  • Figure 3: $K_7\setminus H_6$ (left) and $s_2s_5(K_7\setminus H_6)$ (right)
  • Figure 4: Case 3, red $K_6\setminus C_5$ centered at $v_4$
  • Figure 5: Case 3, blue $K_6\setminus C_5$ on $\{o,w_1,w_2,v_6,v_7,v_1\}$ centered at $v_7$
  • ...and 2 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof : Proof of \ref{['main']}
  • Lemma 4
  • proof
  • Proposition 5: Koizumi
  • proof
  • Proposition 6
  • proof
  • ...and 12 more