Conjecture Suppose runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least (along the track) away from every other runner.
For a finite (additive) abelian group , the Davenport constant of , denoted , is the smallest integer so that every sequence of elements of with length has a nontrivial subsequence which sums to zero.
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.
Problem Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?
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 Is there a minimum integer such that the vertices of any digraph with minimum outdegree can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least ?
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.
Conjecture Let be a Cantor set embedded in . Is there a self-homeomorphism of for every greater than so that moves every point by less than and does not intersect ? Such an embedded Cantor set for which no such exists (for some ) is called "sticky". For what dimensions do sticky Cantor sets exist?
Problem Does the following equality hold for every graph ?
The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number, we minimize the number of pairs of edges that cross.
Problem Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .
Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .
There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .
Problem Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?
1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Question Is the binary affine cube the only 3-connected matroid for which equality holds in the bound where is the circumference (i.e. largest circuit size) of ?
Conjecture Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).
Conjecture There exists a fixed constant (probably suffices) so that every graft with minimum -cut size at least contains a -join packing of size at least .
Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .
Question Does the Blatter-Specker Theorem hold for ternary relations.
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runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least 
(along the track) away from every other runner. 
, the Davenport constant of 
, is the smallest integer 
so that every sequence of elements of 
has a nontrivial subsequence which sums to zero.
.
of the projective plane is octahedral if all lines in 
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
where 
is the number of vertices. What is the distribution of 
for 
? 
such that the vertices of any digraph with minimum outdegree 
can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least 
has a proper 
-
generic matrix? 
for 
. Its determinant is a homogeneous form of degree 
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 
. 
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 
be a 
. Is there a self-homeomorphism 
of 
greater than 
so that 
does not intersect 
do sticky Cantor sets exist? 
contains every oriented tree of order 
then 
for every funcoid 
and 
on the source and destination of 
then 
for every funcoid 
, 
. ![\[ \text{pair-cr}(G) = \text{cr}(G) \]](/files/tex/8cece1e00bb0e9fc122e0a5cad0dab2681cf33a4.png)
of a graph 
, we minimize the number of pairs of edges that cross. 
in 
, let the symmetry group of 
ie: isotopy classes of diffeomorphisms of 
of 
is the unlink. Let 
be the subgroup of 
consisting of diffeomorphisms of 
given by restricting the diffeomorphism to the 
is a signed symmetric group -- the wreath product of a symmetric group with 
. 
-uniform hypergraph on 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
, what is the smallest integer 
such that 
? 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
, let 
be the smallest integer 
such that the symmetric group 
on the set of all words of length 
(
is the shuffle permutation defined by 
, and 
is the exchange group consisting of all permutations in 
letters in the words. 
, for all 
the only 3-connected matroid for which equality holds in the bound 
where 
is the circumference (i.e. largest circuit size) of 
? 
is connected. Then 
are invertible 
matrices with entries in 
for a prime 
, then there is a 
submatrix 
of ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
, every 
vertices has a Hamilton cycle. 
(probably 
suffices) so that every graft with minimum 
-cut size at least 
. 
. 
and 
are 
, then 
the number of structures in 
. For any class 
, the function 
. 
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