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.
Conjecture An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the vertices' trees, the star with vertices has the most endomorphisms, while the path with vertices has the least endomorphisms.
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 ?
Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?
Conjecture For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.
An alternating walk in a digraph is a walk so that the vertex is either the head of both and or the tail of both and for every . A digraph is universal if for every pair of edges , there is an alternating walk containing both and
Question Does there exist a locally finite highly arc transitive digraph which is universal?
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 .
Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
Conjecture For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .
Conjecture Can the approximation ratio be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than -hardness?
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so that every 
, 
so that 
does not contain an odd edge-cut. 
, it is impossible to cover a square of side greater than 
with 
unit squares. 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
where 
is the number of vertices. What is the distribution of 
for 
? 
and fixed colouring 
of 
with 
colours, there exists 
such that every colouring of the edges of 
contains either 
vertices whose edges are coloured with at most 
colours. 
of 
is the minimum number of crossings in all drawings of 
? 
so that the vertex 
is either the head of both 
and 
or the tail of both 
. A digraph is universal if for every pair of edges 
, there is an alternating walk containing both 
and 
-uniform hypergraph on 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
on 
) 
(The join operation is taken on the lattice of filters with reversed order.) ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
and the nonzero derivatives of 
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
, 
, or 
. 
has chromatic number at most 
.
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? ![\[ w(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil. \]](/files/tex/d2391343543ce03d861e6eb2f4985d52e309525d.png)
. 
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