The Hadwiger Number of Jordan Regions is Unbounded

TL;DR

This paper proves that in the plane, there is no upper limit to the number of disjoint translates of a Jordan region that can all touch a common Jordan region, showing the unbounded nature of the Hadwiger number.

Contribution

It demonstrates that the Hadwiger number for Jordan regions in the plane is unbounded, providing a construction for arbitrarily many touching translates.

Findings

Unbounded Hadwiger number for Jordan regions

Existence of arbitrarily large sets of touching translates

Planar topological disks can have arbitrarily many touching copies

Abstract

We show that for every n > 0 there is a planar topological disk A_0 and n translates A_1, A_2, ..., A_n of A_0 such that the interiors of A_0, ... A_n are pairwise disjoint, but with each A_i touching A_0 for 1 <= i <= n.