Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.
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.
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 .
Conjecture For every fixed graph , there exists a constant , so that every graph without an induced subgraph isomorphic to contains either a clique or an independent set of size .
An -factor in a graph is a set of vertex-disjoint copies of covering all vertices of .
Problem Let be a fixed positive real number and a fixed graph. Is it NP-hard to determine whether a graph on vertices and minimum degree contains and -factor?
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?
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is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
structurally stable diffeomorphism is hyperbolic. 
be a class of finite relational structures. We denote by 
the number of structures in 
. For any class 
, the function 
such that 
![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
Square-Free?
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 
. 
, there exists a constant 
, so that every graph 
without an induced subgraph isomorphic to 
. 
for every filter objects 
and 
and a funcoid 
? 
be a fixed positive real number and 
contains and 
denote the graph with vertex set consisting of all lines through the origin in 
and two vertices adjacent in 
? 
so that every 
, every (loopless) graph 
immerses 
. 
. Is there a self-homeomorphism 
of 
greater than 
so that 
does not intersect 
, 
so that 
does not contain an odd edge-cut. 
on an even number of vertices can be decomposed into 
directed paths. 
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