Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?
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 .
Conjecture Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in -outerplanar graphs or tree-width graphs?
Conjecture If is a finite field with at least 4 elements and is an invertible matrix with entries in , then there are column vectors which have no coordinates equal to zero such that .
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
Conjecture There is a function such that for every graph ,
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical sides is the winner. What is the outcome of the game given optimal play? Note: Roland Bacher and Shalom Eliahou proved that every 15 x 15 binary matrix contains four equal entries (all 0's or all 1's) at the vertices of a square with horizontal and vertical sides. So the game must result in a winner (the first player) when n=15.
Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
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.
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inequalities in dimension 
is 
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
be an eulerian graph of minimum degree 
, and let 
be an eulerian tour of 
. 
, there exists a Fibonacci number divisible by 
, 
does not divide 
where 
is the Legendre symbol. 
-spheres bound (rational) homology 
-outerplanar graphs or tree-width graphs? 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to 
is a finite field with at least 4 elements and 
is an invertible 
matrix with entries in 
which have no coordinates equal to zero such that 
. 
of 
and a (non-proper) 
such that edges with the same colour do not cross with respect to 
for some edges 
, then 
and 
receive distinct colours. 
is the minimum integer 
be the graph obtained by subdividing each edge of 
such that for every graph 
with specified vertices 
and 
contains an out-branching rooted at 
? 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
contains 
. 
, whether there exists a homomorphism from 
for some constant 
? 
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. 
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