Venn diagrams and symmetric chain decompositions in the Boolean lattice
Summary: We show that symmetric Venn diagrams for \(n\) sets exist for every prime \(n\), settling an open question. Until this time, \(n=11\) was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice \(\mathcal B_n\), and prove that such decompositions exist for all prime \(n\). A consequence of the approach is a constructive proof that the quotient poset of \(\mathcal B_n\), under the relation ``equivalence under rotation'', has a symmetric chain decomposition whenever \(n\) is prime. We also show how symmetric chain decompositions can be used to construct, for all \(n\), monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.
scientific article; zbMATH DE number 2038348