In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.
Problem Let be a tournament with edges colored from a set of three colors. Is it true that must have either a rainbow directed cycle of length three or a vertex so that every other vertex can be reached from by a monochromatic (directed) path?
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
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 .
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.
Problem The valency-variety of a graph is the number of different degrees in . Is the chromatic number of any graph with at least two vertices greater than
Conjecture For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.
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 be an eulerian graph of minimum degree , and let be an eulerian tour of . Then admits a decomposition into cycles none of which contains two consecutive edges of .
Suppose is a finite group, and is a positive integer dividing . Suppose that has exactly solutions to . Does it follow that these solutions form a subgroup of ?
Question What is the least integer such that every set of at least points in the plane contains collinear points or a subset of points in general position (no three collinear)?
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
If , are graphs, a function is called cycle-continuous if the pre-image of every element of the (binary) cycle space of is a member of the cycle space of .
Problem Does there exist an infinite set of graphs so that there is no cycle continuous mapping between and whenever ?
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be a tournament with edges colored from a set of three colors. Is it true that 
so that every other vertex can be reached from 
be a 
-dimensional smooth submanifold of 
, 
. By the Jordan-Brouwer separation theorem, 
-manifolds which share 
? ie: is 
is a tournament of order 
, then it contains 
arc-disjoint transitive subtournaments of order 3. 
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
-colored countably infinite complete subgraph. Then 
, 
, or 
.
(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
. We say that 
monochromatic empty triangles. 
of a graph 
contains a directed cycle of length at least 
. 
are prime.
-flow for some 
-flow. 
of 
can be represented as the sum of an odd prime number and an odd 
, there is a constant 
(possibly 
) so that the union (as multisets) of any 
contains an additive basis. 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
is an invertible 
matrix, then there is an 
of ![$ [A A] $](/files/tex/d1e9d82c656535b507686183e640178057fae455.png)
so that 
is nonzero. 
are matroids on 
and 
for every partition 
of 
with 
which is independent in every 
. 
be an eulerian tour of 
such that 
or prove that such integers do not exist. 
. Suppose that 
. Does it follow that these solutions form a subgroup of 
such that every set of at least 
are graphs, a function 
is called cycle-continuous if the pre-image of every element of the (binary) cycle space of 
so that there is no cycle continuous mapping between 
and 
whenever 
? 
is not an integer. 
are invertible 
for a prime 
submatrix ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
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