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.
A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .
A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .
Conjecture Let be an integer and a -regular graph. If is a class 1 graph, then .
Problem Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .
Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .
There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .
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.
Problem Let and be two -uniform hypergraph on the same vertex set . Does there always exist a partition of into classes such that for both , at least hyperedges of meet each of the classes ?
Conjecture A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
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.
The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
Conjecture If two graphs on vertices have the same deck, then they are isomorphic.
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 For which values of and are there bi-colored graphs on vertices and different colors with the property that all the monochromatic colorings have unit weight, and every other coloring cancels out?
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be a simple graph with 
vertices and list chromatic number 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
-degenerate if every subgraph of 
.
, the union of any 
-degenerate graph. 
, then 
cycles of length 
. 
-connected finite graph, is there an assignment of lengths 
to the edges of 
-geodesic cycle is 
-flow 
on 
of 
from the edge set of 
, for all 
, and 
. The circular flow number of 
has a nowhere-zero 
, and it is denoted by 
.
be an integer and 
-regular graph. If 
. 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
in 
, let the symmetry group of 
ie: isotopy classes of diffeomorphisms of 
of 
is the unlink. Let 
be the subgroup of 
consisting of diffeomorphisms of 
given by restricting the diffeomorphism to the 
is a signed symmetric group -- the wreath product of a symmetric group with 
. 
-
(mod 
) for every vertex 
. 
always rational? 
exist a 
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)
. 
such that for every hexagonal graph 
, 
, what is the smallest integer 
such that 
? 
be a prime natural number. Find all primes 
, such that 
.
and graph 
. 
people admits a solution is 
. 
and 
be two 
. Does there always exist a partition of 
such that for both 
, at least 
hyperedges of 
meet each of the classes ![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
. 
be a class of finite relational structures. We denote by 
the number of structures in 
. For any class 
, the function 
. 
vertices have the same deck, then they are isomorphic. 
, let 
be the statement that given any exact 
, 
, or 
. 
are there bi-colored graphs on 
for reloid 
. 
for every funcoid 
(see below mentioned online article).
has distinct subset sums if distinct subsets of 
have distinct sums.
whenever 
has distinct subset sums. 
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