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.
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 .
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.
Conjecture Let be a field of characteristic zero. A collection of polynomials in variables defines an automorphism of if and only if the Jacobian matrix is a nonzero constant.
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .
The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.
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 .
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 .
The star chromatic index of a graph is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.
Question Is it true that for every (sub)cubic graph , we have ?
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.
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
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be a set of points in the plane. Two points 
and 
in 
there is an integer 
such that every set of at least 
collinear points or 
pairwise visible points. 
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 
in 
such that the fundamental group of the complement has an unsolvable word problem? 
generators and exponent 
and 
of polynomials in variables 
defines an automorphism of 
if and only if the Jacobian matrix is a nonzero constant. 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
in the hypercube. 
with the property that 
for every 
with odd size.
for every 
edges whose deletion destroys every triangle. 
. 
appears once in Pascal's triangle, 
appears twice, 
appears three times, and 
appears 
times. There are infinite families of numbers known to appear 
times is 
. It is not known whether any number appears more than 
. See 
, 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 
. 
, 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 
.
of a graph 
? 
contains every oriented tree of order 
, there is a constant 
(possibly 
) so that the union (as multisets) of any 
contains an additive basis. 
, every 
-diregular oriented graph on at most 
vertices has a Hamilton cycle. 
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
-minor-free graph is 
-list-colourable for some constant 
. 
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