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.
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?
Conjecture Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Question What is the Waring rank of the determinant of a generic matrix?
For simplicity say we work over the complex numbers. The generic matrix is the matrix with entries for . Its determinant is a homogeneous form of degree , in variables. If is a homogeneous form of degree , a power sum expression for is an expression of the form , the (homogeneous) linear forms. The Waring rank of is the least number of terms in any power sum expression for . For example, the expression means that has Waring rank (it can't be less than , as ).
The generic determinant (or ) has Waring rank . The Waring rank of the generic determinant is at least and no more than , see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.
Conjecture Suppose that is a -edge-critical graph. Suppose that for each edge of , there is a list of colors. Then is -edge-colorable unless all lists are equal to each other.
Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?
A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Problem Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.
Problem has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .
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 If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
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![\[ F_n = 2^{2^n } + 1 \]](/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.
? 
exist a 
so that the power series 
is bounded for all 
. 
and 
are there bi-colored graphs on 
is a graph and ![$ p \in [0,1] $](/files/tex/1d076f7332523eb59bd7deae19667f83f0a3b6e0.png)
, we let 
denote a subgraph of 
.
so that 
? 
-connected cubic graph and let 
be a 
is connected. Then 
into 
so that the graph induced by 
is acyclic for 
. 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
generic matrix? 
for 
. Its determinant is a homogeneous form of degree 
variables. If 
is a homogeneous form of degree 
, the 
(homogeneous) linear forms. The Waring rank of 
in any power sum expression for 
means that 
has Waring rank 
).
generic determinant 
(or 
) has Waring rank 
. The Waring rank of the 
generic determinant is at least 
and no more than 
, see for instance 
, we write 
if 
satisfies 
. The problem of finding such an integer 
(with 
) is the Discrete Log Problem.
-edge-critical graph. Suppose that for each edge 
of 
of 
-edge-colorable unless all lists are equal to each other. 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [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?
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
covering design, or covering, is a family of 
-subsets, called blocks, chosen from a 
-set, such that each 
-subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by 
.
. Find a procedure for constructing minimal coverings. 
has the homotopy-type of a product space 
where 
is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of 
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 
? 
, then there is a 
such that 
meets 
in at most one vertex, 
. 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
such that its fundamental group has an element of finite order?
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