Venn Diagrams with Few Vertices
Bette Bultena, Frank Ruskey · The Electronic Journal of Combinatorics · 1998
An $n$-Venn diagram is a collection of $n$ finitely-intersecting simple closed curves in the plane, such that each of the $2^n$ sets $X_1 \cap X_2 \cap \cdots \cap X_n$, where each $X_i$ is the open interior or exterior of the $i$-th curve, is a non-empty connected region. The weight of a region is the number of curves that contain it. A region of weight $k$ is a $k$-region. A monotone Venn diagram with $n$ curves has the property that every $k$-region, where $0 1$.