On Double-k-Systems of Spheres

Abstract

An oriented sphere in ${\mathbb {R}}^n$ is a sphere for which one side (inside or outside) is distinguished as its positive side. Two oriented spheres are said to be properly tangent if they are tangent and the positive side of one sphere is contained in the positive side of the other. A set of 2k spheres $\{\alpha _1,\dots ,\alpha _k,\beta _1,\dots ,\beta _k\}$ in ${\mathbb {R}}^n$ is called a double-k-system in ${\mathbb {R}}^n$ if $\alpha _i$ and $\beta _j$ are properly tangent for $i\ne j$, and no other pair in the set is properly tangent. It is known that if a double-k-system exists in ${\mathbb {R}}^n$ then $k\le n+3$, and that for every n there is a double-$(n+2)$-system in ${\mathbb {R}}^n$. Moreover, in ${\mathbb {R}}^3$ there is a double-6-system, and in ${\mathbb {R}}^1$ no double-4-system exists. In this paper, we show that there is no double-5-system in the plane, which answers a problem posed in Maehara and Tokushige (Eur J Comb 30:1337–1351, 2009). We also derive a few related results in dimensions $n>3$.