Conjecture Let be an eulerian graph of minimum degree , and let be an eulerian tour of . Then admits a decomposition into cycles none of which contains two consecutive edges of .
Conjecture Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .
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.
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.
Conjecture If is a cubic graph not containing a triangle, then it is possible to color the edges of by five colors, so that the complement of every color class is a bipartite graph.
Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .
Conjecture If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
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 .
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
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 .
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is 2-large, then 
be an eulerian graph of minimum degree 
, and let 
be an eulerian tour of 
, 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)
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. 
, where 
is some ordinal. This 
as 
.
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)
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.) 
are invertible 
matrices with entries in 
for a prime 
, then there is a 
submatrix ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
such that for every hexagonal graph 
, 
the following inequality holds 
![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
Square-Free?
, the number of even latin squares of order 
, it is impossible to cover a square of side greater than 
unit squares. 
, is the smallest integer 
so that every sequence of elements of 
has a nontrivial subsequence which sums to zero.
? 
be abelian groups and let 
and 
satisfy 
and 
. If there is a 
to 
, then every graph with a B-flow has a B'-flow. 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
is a completary multifuncoid of the form 
, then 
is a completary multifuncoid of the form 
. 
it holds
if 
if 
; \item 
; \item 
. 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
, that every sufficiently large 
vertices whose deletion results in a 
there is an integer 
such that every digraph of minimum outdegree at least 
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