Conjecture For all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order .
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 .
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 ?
Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
\item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.
See details, exact definitions, and attempted proofs here.
For a graph , let denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let denote the cardinality of a minimum feedback vertex set (set of vertices so that is acyclic).
Conjecture Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
Conjecture Let be a -separable compact reflexive symmetric funcoid and be a reloid such that \item ; \item .
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
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and 
, the graph 
has equivalence covering number 
. 
can be represented as the sum of an odd prime number and an odd 
is a graph, 
is admissable, and 
is even for every 
, then 
has a faithful cover. 
such that every digraph of minimum outdegree at least 
has chromatic number at most 
.
be a finite graph, let 
, and let 
be the edge set of a forest chosen uniformly at random from all forests of ![\[ {\mathbb P}(e \in F \mid f \in F}) \le {\mathbb P}(e \in F) \]](/files/tex/1fe7ac9c579238670e3aa16ae401e8194e1c3da3.png)
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 
. 
hold over graphs of bounded tree-width? \item Is 
included in 
over graphs? \item Does 
exist a 
denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let 
denote the cardinality of a minimum feedback vertex set (set of vertices 
so that 
is acyclic). 
. 
contains every 2-regular graph on 
vertices as a minor. 
is a 
-separable (the same as 
for symmetric transitive) compact funcoid and 
is a uniform space (reflexive, symmetric, and transitive endoreloid) such that 
. Then 
. 
. 
. 
such that its fundamental group has an element of finite order?
cycles. 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
on an even number of vertices can be decomposed into 
directed paths. 
-flow 
on 
from the edge set of 
, for all 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
is a 2-transition system for 
has a compaible decomposition. 
. Suppose that 
and each vertex of 
vertices of 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
so that all pairwise distances between points in 
are rational? 
, there is an integer 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to 
is for every sets 
and 
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