Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
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 ?
Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .
Conjecture For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.
Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.
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 .
A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
Problem Is it true that for every , all but finitely many -regular graphs have friendly partitions?
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 ?
Conjecture The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.
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complex numbers 
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
is a finite 
-regular graph, where 
, then 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
using 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
be a set of points in the plane. Two points 
and 
in 
there is an integer 
collinear points or 
pairwise visible points. 
in 
such that the fundamental group of the complement has an unsolvable word problem? 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
be an integer and let 
denote the least integer 
, for 
a large prime, always contain a rainbow 
if each of the color classes is of size of either 
or 
? 
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 
. Suppose that 
. Does it follow that these solutions form a subgroup of 
points in the plane in general position contains a subset of 
-connected finite graph, is there an assignment of lengths 
to the edges of 
? 
denote the graph with vertex set consisting of all lines through the origin in 
and two vertices adjacent in 
? 
contains a directed cycle of length at least 
. 
is 2-large, then 
, then 
cycles of length 
. 
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