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?
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 .
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.
For any simple digraph , we let be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and be the size of the smallest feedback edge set.
Conjecture If is a simple digraph without directed cycles of length , then .
Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .
A permutation is called a Roller Coaster permutation if . Let be the set of all Roller Coaster permutations in .
Conjecture For ,
\item If , then . \item If , then with .
Conjecture (Odd Sum conjecture) Given ,
\item If , then is odd for . \item If , then for all .
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
If is a finite set of points which is 2-colored, an empty triangle is a set with so that the convex hull of is disjoint from . We say that is monochromatic if all points in are the same color.
Conjecture There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.
Conjecture \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.
Conjecture Denote by the number of non-Hamiltonian 3-regular graphs of size , and similarly denote by the number of non-Hamiltonian 3-regular 1-connected graphs of size .
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 .
Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .
Question Does the Blatter-Specker Theorem hold for ternary relations.
Conjecture Let be the disjoint union of the graphs and and let be a surface. Is it true that every optimal drawing of on has the property that and are disjoint?
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, determine whether or not 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
is a prime iff 
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 
and external set 
). Then there exist a multifuncoid 
of the form 
such that 
regarding the filtrator 
. 
, is the smallest integer 
so that every sequence of elements of 
has a nontrivial subsequence which sums to zero.
be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and 
be the size of the smallest 
, then 
. 
denote the set of all permutations of ![$ [n]=\set{1,2,\ldots,n} $](/files/tex/470bd09059264370b76b4da90b77dc370c7e0e7c.png)
. Let 
and 
denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in 
. Let 
denote the set of subsequences of 
denote 
.
is called a Roller Coaster permutation if 
. Let 
be the set of all Roller Coaster permutations in 
,
, then 
. \item If 
, then 
with 
. 
,
is odd for 
. \item If 
for all 
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
is 2-large, then 
? 
, such that 
.
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
-regular graph, where 
, then 
vertices has an edge colouring such that:
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
is disjoint from 
. We say that 
with the following property. If 
monochromatic empty triangles. ![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
of a locally finite graph ![\[ w(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\rceil. \]](/files/tex/d2391343543ce03d861e6eb2f4985d52e309525d.png)
. 
the number of non-Hamiltonian 3-regular graphs of size 
, and similarly denote by 
the number of non-Hamiltonian 3-regular 1-connected graphs of size 
? 
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 
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
is at most 
.
be a class of finite relational structures. We denote by 
the number of structures in 
. For any class 
, the function 
. 
and 
and let 
be a surface. Is it true that every optimal drawing of 
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