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.
Conjecture Suppose runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least (along the track) away from every other runner.
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.
We say that a set is -universal if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.
Question Does there exist an -universal set of size ?
Conjecture Let be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists such that is an element of at least half the members of .
Conjecture Is it possible to color edges of the complete graph using colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
Equivalently: is the star chromatic index of linear in ?
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 Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.
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 .
Problem Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?
Conjecture If is a simple graph which is the union of pairwise edge-disjoint complete graphs, each of which has vertices, then the chromatic number of is .
Conjecture Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
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then 
for every funcoid 
and atomic f.o. 
and 
on the source and destination of 
then 
for every funcoid 
, 
. 
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 
vertices whose edges are coloured with at most 
colours. 
(along the track) away from every other runner. 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
.
of the projective plane is octahedral if all lines in 
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
? 
-graph is an 
with the property that 
for every 
with odd size.
for every 
is 
-universal if every 
, and all edges are (non-intersecting) straight line segments. 
? 
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists 
such that 
-
(mod 
) for every vertex 
. 
there is an integer 
such that every set of at least 
collinear points or an empty hexagon. ![\[ F_n = 2^{2^n } + 1 \]](/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.
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
such that for any pair 
there are 
edge-disjoint paths from 
to 
. 
such that the list chromatic number of any bipartite graph 
is at most 
.
can be represented as the sum of an odd prime number and an odd 
is a graph, 
is admissable, and 
is even for every 
, then 
has a faithful cover. 
, let 
be the statement that given any exact 
, 
, or 
. 
of co-prime positive integers 
for 
. 
be natural numbers with 
. It follows from the pigeon-hole principle that there exist distinct subsets 
with 
. Is it possible to find such a pair 
in polynomial time? 
be a circuit in a bridgeless cubic graph 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
, that every sufficiently large 
vertices whose deletion results in a 
be an integer and let 
be a minor minimal 
. Then either 
minor for some 
or 
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