Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

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