THE ELECTRONIC JOURNAL OF COMBINATORICS 4 (1997), DS #5.

# Venn Diagram Survey Open Problems

## Open problems related to Venn diagrams.

• Find a symmetric Venn diagram for n=11, or prove that no such diagram exists.
• How many symmetric Venn diagrams are there for n=7? How many of these are simple?
• Is it true that every simple Venn diagram of n curves can be extended to a simple Venn diagram of n+1 curves by the addition of a suitable curve? [That this is true is a conjecture of Winkler [Wi]. This was proven to be true for not necessarily simple Venn diagrams by Chilakamarri, Hamburger, and Pippert [CHP96].] Equivalently: Is every planar dual graph of a simple Venn diagram Hamiltonian?
• Find a 6-Venn diagram in which each curve is a triangle. [There is a simple symmetric 5-Venn diagram in which each curve is a triangle; see Grunbaum and Winkler [GW].] Find a 6-Venn diagram in which each curve is a rectangle. These problems are both from Grünbaum [Gr84b].
• Find a 6-Venn diagram made from equilateral triangles. Even the problem of finding a 6-Independent family made from equilateral triangles is open. This problem from Grünbaum [Gr75].
• Find a Brunnian link whose minimal projection is a symmetric Venn diagram of order 7 or prove that no such link exists.
• Find a symmetric k-fold (k > 1) Venn diagram on the sphere. Note the such a diagram cannot exist on the plane since there is only one unbounded region.

 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 (1997), DS #5.