Find a symmetric Venn diagram for n = 13, or prove that
no such diagram exists.
How many symmetric Venn diagrams are there for n = 5?
There is only one symmetric simple 5-Venn diagram.
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 rectangle.
This problem is from Grünbaum
[Gr84b].
His other problem of finding a 6-Venn diagram in which each
curve is a triangle has been answered affirmatively.
See the six amazing triangles.
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].
There is no Venn diagram of 8 convex quadrangles.
Does there exist a Venn diagram of 8 quadrangles not required
to be convex, or of 8 pentagons?
These problems are due to Grünbaum.
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 > 2) Venn diagram on
the sphere or prove that no such diagram exists.
For n = 2 there are
diagrams that, in the plane, have
rotational symmetry, and and examples that have
mirror-image symmetry (these examples
due to Branko Grünbaum).
For n > 2, find a "symmetric" diagram on the sphere where the
group of symmetries is different than those used in the section on
symmetric Venn diagrams.
In that section
the group of symmetries consisted of rotations about a single axis,
or rotations and a flip about that axis.
In other words, the problem is:
Are there other groups of isometries that act transitively
on the curves of some Venn diagram on the sphere?
The least number of vertices in a monotone Venn diagram is known to be
C(n,n/2); what is the least number of vertices in a general
Venn diagram?
The lower bound of
ceiling((2^{n}-2)/(n-1))
is known to be achieveable for
n < 8, so the first open case is n = 8.
It is not known whether there is a rigid minimum vertex Venn diagram
for n = 7.
Are there rigid^{*}
monotone n-Venn diagrams with
C(n,n/2) vertices?
The constructions of [BGR]
have C(n,n/2) vertices but have many separable
vertices.
The dual of a simple Venn diagram is a maximal planar spanning
subgraph of the hypercube.
Does every maximal spanning subgraph of the hypercube occur as the
Venn graph of some Venn diagram?
For every n, is there a convex Venn diagram made from congruent
curves?
Over all n-Venn diagrams, which requires the greatest number of
straight line segments to draw it?