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 .
Let be a non-empty finite set. Given a partition of , the stabilizer of , denoted , is the group formed by all permutations of preserving each block of .
Problem () Find a sufficient condition for a sequence of partitions of to be complete, i.e. such that the product of their stabilizers is equal to the whole symmetric group on . In particular, what about completeness of the sequence , given a partition of and a permutation of ?
Conjecture (Beneš) Let be a uniform partition of and be a permutation of such that . Suppose that the set is transitive, for some integer . Then
Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
\item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.
See details, exact definitions, and attempted proofs here.
Conjecture For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.
Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?
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.
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.
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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?
(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
so that all pairwise distances between points in 
are rational? 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
.
contains every 2-regular graph on 
vertices as a minor. 
be a non-empty finite set. Given a partition 
of 
, is the group formed by all permutations of 
.
) Find a sufficient condition for a sequence of partitions 
of 
is equal to the whole symmetric group 
on 
, given a partition 
of 
be a uniform partition of 
be a permutation of 
. Suppose that the set 
is transitive, for some integer 
. Then 
is for every sets 
and 
:
? 
and fixed colouring 
of 
with 
colours, there exists 
such that every colouring of the edges of 
contains either 
vertices whose edges are coloured with at most 
colours. 
is a non-empty graph containing no induced odd cycle of length at least 
, then there is a 
-vertex colouring of 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. 
and every positive integer 
so that every simple 
-regular graph 
has a 
with 
. 
paths. 
, 
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
is a 2-transition system for 
has a compaible decomposition. 
such that 
or prove that such integers do not exist. ![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
Square-Free?
, determine whether or not 
so that the power series 
is bounded for all 
. 
is not an integer. 
covering design, or covering, is a family of 
-set, such that each 
.
. Find a procedure for constructing minimal coverings. 
for some constant 
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