Conjecture Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition It is easy to prove that is the infinitely small right neighborhood filter of point .
If proved true, the conjecture then can be generalized to a wider class of posets.
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 There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .
The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.
Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
Note that the above is a generalization of monotone Galois connections (with and replaced with suprema and infima).
Then we have the following diagram:
What is at the node "other" in the diagram is unknown.
Conjecture "Other" is .
Question What repeated applying of and to "other" leads to? Particularly, does repeated applying and/or to the node "other" lead to finite or infinite sets?
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.
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be the complete funcoid corresponding to the usual topology on extended real line ![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
. Let 
be the order on this set. Then 
is a complete funcoid. 
is the infinitely small right neighborhood filter of point ![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
. 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. 
be an 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
for every endo-
? 
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
-colored countably infinite complete subgraph. Then 
, 
, or 
.
. 
appears once in Pascal's triangle, 
appears twice, 
appears three times, and 
appears 
times. There are infinite families of numbers known to appear 
times is 
. It is not known whether any number appears more than 
. See 
be an integer and let 
denote the least integer 
such that there exists a simple graph on 
spanning trees. Then 
denote the 
? 
paths. 
we have 
. Then 
is graphic. 
of 
with specified vertices 
and 
contains an out-branching rooted at 
, every 
-diregular oriented graph on at most 
vertices has a Hamilton cycle. 
? 
. 
be a 
, 
. By the Jordan-Brouwer separation theorem, 
? ie: is 
, it is impossible to cover a square of side greater than 
unit squares. ![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
is for every sets 
and 
such that the list chromatic number of any bipartite graph 
is at most 
.
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
.
of the projective plane is octahedral if all lines in 
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
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