Planar soap bubble froths have bubbles that meet in at most three at a point whereas in dimension three typically at most four bubbles meet.
In this paper it is shown that it is not possible to decompose the \(n\)-dimensional space \({\mathbb R}^n\) into sets of bounded diameter where at most \(n\) such sets meet. The base is Lebesgue’s covering theorem which says that it is impossible to decompose the unit cube in \({\mathbb R}^n\) into pieces of diameter less than 1 so that at most \(n\) pieces meet at a point. Among other corollaries the authors prove the following: There is no decomposition of the unit \(n\)-ball of \({\mathbb R}^n\) into sets of diameter less than \(\frac{2}{\sqrt{n}}\) that meet at most \(n\) at a point. Consequently, if a 3-dimensional soap bubble cluster covers a ball of diameter at least \(\frac{\sqrt{3}}{2}\) times the largest diameter of a bubble in the cluster, then there must be a point where four bubbles meet.