Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice Preliminary Draft

In this paper 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. We show that the problem can be reduced to flnding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice Bn, and prove that such decompositions exist for all prime n. A consequence of the construction is that the quotient poset of Bn, under the relation \equivalence under rotation", has a symmetric chain decomposition whenever n is prime. We also show how SCDs can be used to construct, for all n, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.