On the Number of Mutually Touching Cylinders
András Bezdek · 2005
In a three-dimensional arrangement of 25 congruent nonoverlapping infinite circular cylinders there are always two that do not touch each other. 1. Introduction The following problem was posed by Littlewood [1968]: What is the maximum number of congruent infinite circular cylinders that can be arranged in ℝ 3 so that any two of them are touching? Is it 7? This problem is still open. The analogous problem concerning circular cylinders of finite length became known as a mathematical puzzle due to a the popular book [Gardner 1959]: Find an arrangement of 7 cigarettes so that any two touch each other. The question whether 7 is the largest such number is open. For constructions and for a more detailed account on both of these problems see the research problem collection [Moser and Pach ≥ 2005]. A very large bound for the maximal number of cylinders in Littlewood's original problem was found by the author in 1981 (an outline proof was presented at the Discrete Geometry meeting in Oberwolfach in that year). The bound was expressed in terms of various Ramsey constants, and so large that it merely showed the existence of a finite bound. In this paper we use a different approach to show that at most 24 cylinders can be arranged so that any two of them are touching: THEOREM 1. In an arrangement of 25 congruent nonoverlaping infinite circular cylinders there are always two that do not touch each other.