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 .
Conjecture Let be the open unit disk in the complex plane and let be open sets such that . Suppose there are injective holomorphic functions such that for the differentials we have on any intersection . Then those differentials glue together to a meromorphic 1-form on .
Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
Conjecture Every arrangement graph of a set of great circles is -colourable.
Conjecture Can the approximation ratio be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than -hardness?
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 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.
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be a graph and let 
such that for any pair 
there are 
edge-disjoint paths from 
to 
in 
edge-disjoint trees, each of which contains 
. 
be the open unit disk in the complex plane and let 
be open sets such that 
. Suppose there are injective holomorphic functions 
such that for the differentials we have 
on any intersection 
. Then those differentials glue together to a meromorphic 1-form on 
of co-prime positive integers 
for 
. 
with 
so that 
has a 
. 
of great circles on a sphere with no three circles meeting at a point. The arrangement graph of 
-colourable. 
. The representation function 
for 
is given by the rule 
. We call 
is never 
.
hyperplanes in dimension 
is not greater than 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
for every endo-
? 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
are independent random variables with ![$ \mathbb{E}[X_i] \leq \mu $](/files/tex/e0268221532981debea25e9446c8ee6f112e1881.png)
, then ![$$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$](/files/tex/03dc1130142ee6fefcc33888e2fb6137211bf327.png)
. 
is 
? 
is a graph, 
is admissable, and 
is even for every 
, then 
has a faithful cover. 
to the edges of 
-geodesic cycle is 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
is defined to be the mimic number of any positive integer 
. 
is a 
for all primes 
, where 
is prime. 
such that 
meets 
in at most one vertex, 
. 
, then there is a 
-vertex colouring of 
paths. 
be an integer and let 
denote the least integer 
has distinct subset sums if distinct subsets of 
so that 
whenever 
has distinct subset sums. 
in the hypercube. 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
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