We say that a set is -universal if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.
Question Does there exist an -universal set of size ?
Question What is the Waring rank of the determinant of a generic matrix?
For simplicity say we work over the complex numbers. The generic matrix is the matrix with entries for . Its determinant is a homogeneous form of degree , in variables. If is a homogeneous form of degree , a power sum expression for is an expression of the form , the (homogeneous) linear forms. The Waring rank of is the least number of terms in any power sum expression for . For example, the expression means that has Waring rank (it can't be less than , as ).
The generic determinant (or ) has Waring rank . The Waring rank of the generic determinant is at least and no more than , see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
Problem has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .
Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.
Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.
In particular, this implies:
Conjecture Twin Prime Conjecture: There are an infinite number of twin primes.
Question \item Does hold over graphs of bounded tree-width? \item Is included in over graphs? \item Does have a 0-1 law? \item Are properties of Hanf-local? \item Is there a logic (with an effective syntax) that captures ?
Problem Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .
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is 
-universal if every 
, and all edges are (non-intersecting) straight line segments. 
? 
generic matrix? 
for 
. Its determinant is a homogeneous form of degree 
, in 
variables. If 
is a homogeneous form of degree 
, the 
(homogeneous) linear forms. The Waring rank of 
in any power sum expression for 
means that 
has Waring rank 
(it can't be less than 
).
generic determinant 
(or 
) has Waring rank 
. The Waring rank of the 
generic determinant is at least 
and no more than 
, see for instance 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. ![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
Square-Free?
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
be a 
-dimensional smooth submanifold of 
, 
. By the Jordan-Brouwer separation theorem, 
? ie: is 
has the homotopy-type of a product space 
where 
is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of 
such that every graph with average degree smaller than 
for principal funcoid 
and a set 
of funcoids of appropriate sources and destinations. 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
. 
is a 
-chromatic graph on at most 
vertices, then 
. 
be an indexed family of sets.
for 
.
on the lattice 
of all finite unions of products.
a bijection from hyperfuncoids 
defining the inverse bijection? If not, characterize the image of the function 
of complements of elements of the set 
, then 
cycles of length 
. 
hold over graphs of bounded tree-width? \item Is 
included in 
over graphs? \item Does 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
defined inductively by 
, 
, 
, and 
? 
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. 
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