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].
It was proven to be true for not necessarily simple Venn diagrams by
Chilakamarri, Hamburger, and Pippert
[CHP96a].]
Equivalently: Is every planar dual graph of a simple Venn diagram
Hamiltonian?
Find a simple symmetric 11-Venn diagram.
More generally, find a general construction for simple symmetric Venn diagrams.
Find a minimum-vertex symmetric 11-Venn diagram.
Such a Venn diagram would have 209 vertices.
Find a polar symmetric 11-Venn diagram.
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?
It is known that there are at least 33 simple symmetric Venn diagrams.
Find a Brunnian link whose minimal projection is a symmetric
Venn diagram of order 7 or prove that no such link exists.
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[ (2n-2)/(n-1) ]
is known to be achievable 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.
Find a construction for rigid*
monotone n-Venn diagrams with C(n,n/2) vertices.
The constructions of [BGR]
and [GKS]
have C(n,n/2) vertices but have many separable
vertices. Examples for n=7 are known.
The dual of a simple Venn diagram is a maximal planar spanning
subgraph of the hypercube. (The dual is still maximal if the Venn diagram is only rigid.)
Does every maximal planar 3-connected spanning subgraph of the hypercube, where
all faces are quadrilaterals, occur as the dual graph of some simple Venn diagram?
Geometric Problems
For every n, is there a convex Venn diagram made from congruent curves?
That this is possible if the curves are allowed to intersect in
segments was shown in the section on
infinitely intersecting Venn diagrams.
This is known to be true for primes by the [GKS] construction and for
n = 4 (which can be done with congruent ellipses), but is open for
other values of n.
Find a general construction for minimum-area "Venn" diagrams made from polyominoes.
The largest value of n for which a poly-Venn is known to exist is n = 7.
Find a 6-Venn diagram in which each curve is a rectangle;
this problem is from Grünbaum
[Gr84b]. Slightly easier, find a
6-Venn diagram in which each curve is a trapezoid.
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 is from Grünbaum
[Gr75].
As shown on this page, there is no
simple Venn diagram of 8 convex quadrilaterals.
Does there exist a Venn diagram of 8 quadrilaterals not required
to be convex or simple, or of 8 pentagons?
These problems are due to Grünbaum.
As a function of n, what is the maximum value, taken over all simple
convex n-Venn diagrams, of the minimum k necessary to draw
the diagram as a collection of convex k-gons?
For example, for n=7 we know that there exists a simple diagram
(namely, Victoria)
that is drawable with convex 4-gons. However, we suspect that other diagrams are not
drawable with 4-gons, but since they are convex they are drawable by k-gons,
for some k < 126. What is the maximum k over all
7-Venn diagrams? More generally what is the value of k for
n-Venn diagrams? By the results of
[BGR], any convex Venn diagram
can be drawn with (2n-2)-gons.
Can every convex 3-Venn diagram be drawn with ellipses or triangles?
Over all n-Venn diagrams, which requires the greatest number of
straight line segments to draw it?
Generalizations of Venn Diagrams
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 examples that have
mirror-image symmetry (these examples
due to Branko Grünbaum [FGK]).
Prove that there do not exist simple symmetric
independent families
or simple symmetric near-Venn diagrams attaining the bounds given on the
Generalizations and Extensions
page (the numbers I(n) and E(n) give these bounds).
Generalize the result of [GKS] (described
on this page) to find a general
construction for symmetric independent families and/or
near-Venn diagrams meeting the bounds given on the
Generalizations and Extensions
page. Jiang [Ji] gave a construction
for n< 16.
Given a set system, what is the complexity of determining whether it
can be rendered as an Euler diagram?
For a non-standard definition of Venn diagram an NP-completeness result
is known (Johnson and Pollack [JP]).
