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 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.
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?
Let be a non-empty finite set. Given a partition of , the stabilizer of , denoted , is the group formed by all permutations of preserving each block of .
Problem () Find a sufficient condition for a sequence of partitions of to be complete, i.e. such that the product of their stabilizers is equal to the whole symmetric group on . In particular, what about completeness of the sequence , given a partition of and a permutation of ?
Conjecture (Beneš) Let be a uniform partition of and be a permutation of such that . Suppose that the set is transitive, for some integer . Then
Question What is the least integer such that every set of at least points in the plane contains collinear points or a subset of points in general position (no three collinear)?
Conjecture If is a bridgelesscubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .
Conjecture Suppose that is a -edge-critical graph. Suppose that for each edge of , there is a list of colors. Then is -edge-colorable unless all lists are equal to each other.
Conjecture There exists a fixed constant so that every abelian group has a subset with so that the Cayley graph has no clique or independent set of size .
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has a homomorphism to 
. 
are invertible 
matrices with entries in 
for a prime 
, then there is a 
submatrix 
of ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
for every filter objects 
and 
and a funcoid 
? 
of 
is the minimum number of crossings in all drawings of 
and 
are 
has a proper 
-
vertices and list chromatic number 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
let 
. 
be a matroid on 
, let 
be a map, put 
and 
. Then 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
be an integer and let 
denote the least integer 
such that there exists a simple graph on 
. Does the fragment 
have the finite model property? 
which cannot be expressed as a union of 
triangle free graphs? 
, there exist positive integers 
, 
, 
such that 
.
of 
, is the group formed by all permutations of 
.
) Find a sufficient condition for a sequence of partitions 
of 
is equal to the whole symmetric group 
on 
, given a partition 
of 
be a uniform partition of 
be a permutation of 
. Suppose that the set 
is transitive, for some integer 
. Then 
and 
it holds
if 
if 
; \item 
; \item 
. 
and 
are 
, then 
does there exists a 
such that every set of at least 
. 
are independent random variables with ![$ \mathbb{E}[X_i] \leq \mu $](/files/tex/e0268221532981debea25e9446c8ee6f112e1881.png)
, then ![$$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$](/files/tex/03dc1130142ee6fefcc33888e2fb6137211bf327.png)
of 
of 
-edge-colorable unless all lists are equal to each other. 
edges whose deletion destroys every triangle. 
denote the 
? 
so that every abelian group 
so that the 
has no clique or independent set of size 
. 
are simple finite graphs, then 
. 
is the 
.
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