Irreducible uniform cover
= Irreducible uniform cover
A uniform cover is irreducible when it is not the disjoint union, as a multiset, of two nonempty uniform covers. There are only finitely many irreducible uniform covers of a fixed finite set: their multiplicity vectors form an <antichain> in $\mathbb N^{2^n}$, and <Dickson lemma> forbids an infinite antichain there.