Conjecture For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
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 be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
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 .
A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
Problem Is it true that for every , all but finitely many -regular graphs have friendly partitions?
Conjecture Let is a family of multifuncoids such that each is of the form where is an index set for every and is a set for every . Let every for some multifuncoid of the form regarding the filtrator . Let is a graph-composition of (regarding some partition and external set ). Then there exist a multifuncoid of the form such that regarding the filtrator .
Conjecture Let be the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space
This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .
An oriented colouring of an oriented graph is assignment of colours to the vertices such that no two arcs receive ordered pairs of colours and . It is equivalent to a homomorphism of the digraph onto some tournament of order .
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 Let and . Then for any neighborhood there is such that is periodic point of
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for
Conjecture If is a simple graph which is the union of pairwise edge-disjoint complete graphs, each of which has vertices, then the chromatic number of is .
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of 
points in the plane, not all collinear, there is a point in 
lines determined by 
. ![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
has a cycle with length at most 
be a graph, 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
-regular directed graph with no parallel arcs, then 
arc-disjoint directed cycles. 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
and 
are 
is a family of multifuncoids such that each 
is of the form 
where 
is an index set for every 
and 
is a set for every 
. Let every 
for some multifuncoid 
of the form 
regarding the filtrator 
. Let 
is a graph-composition of 
). Then there exist a multifuncoid 
of the form 
such that 
regarding the filtrator 
. 
-size circuits compute the function 
on 
defined inductively by 
, 
, 
, and 
? 
be the space of 
Diffeomorphisms on the connected , compact and boundaryles manifold M and 
the space of 
(
) such that 
exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
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.) 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
and 
. Then for any neighborhood 
there is 
such that 
is periodic point of 
. In the case of Flows this has been solved by Sushei Hayahshy for 
satisfy 
. Then the elements of 
may be ordered 
and 
so that the sums 
are pairwise distinct. ![$ p \in [0,1] $](/files/tex/1d076f7332523eb59bd7deae19667f83f0a3b6e0.png)
, we let 
denote a subgraph of 
? 
if 
if 
; \item 
; \item 
. 
of co-prime positive integers 
for 
. 
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