The Hadwiger number of Jordan regions is unbounded (Q2369936)

The authors establish the following surprising property of 2-dimensional topological disks. For any \(n\geq 1\) there is such a disk \(A\) and \(n\) translates \(A_1,A_2,\dots,A_n\) of it such that the interiors of the translates are pairwise disjoint but each \(A_i\) touches \(A\) for \(1\leq i\leq n\). The authors construct explicitly such a disk \(A\).