Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.
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 Let be a sequence of points in with the property that for every , the points are distinct, lie on a unique sphere, and further, is the center of this sphere. If this sequence is periodic, must its period be ?
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 .
Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
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 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 Let be a Cantor set embedded in . Is there a self-homeomorphism of for every greater than so that moves every point by less than and does not intersect ? Such an embedded Cantor set for which no such exists (for some ) is called "sticky". For what dimensions do sticky Cantor sets exist?
Conjecture For every rational and every rational , there is no polynomial-time algorithm for the following problem.
Given is a 3SAT (3CNF) formula on variables, for some , and clauses drawn uniformly at random from the set of formulas on variables. Return with probability at least 0.5 (over the instances) that is typical without returning typical for any instance with at least simultaneously satisfiable clauses.
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 ?
Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?
For any simple digraph , we let be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and be the size of the smallest feedback edge set.
Conjecture If is a simple digraph without directed cycles of length , then .
Problem Let be an indexed family of filters on sets. Which of the below items are always pairwise equal?
1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
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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.
? 
is a 
for all primes 
, where 
is prime. 
be a sequence of points in 
with the property that for every 
, the points 
are distinct, lie on a unique sphere, and further, 
is the center of this sphere. If this sequence is periodic, must its period be 
? 
be the flow polynomial of a graph 
, the value 
equals the number of 
for every 2-edge-connected graph 
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 
. 
. Does the fragment 
have the finite model property? 
rectangular 
-dimensional boxes each of which has side lengths 
inside an 
? 
and a function 
such that if 
, then every saturated 
has cardinality at least 
? 
of polynomials in variables 
defines an automorphism of 
if and only if the Jacobian matrix is a nonzero constant. 
using 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
be a 
. Is there a self-homeomorphism 
of 
greater than 
so that 
does not intersect 
it holds
if 
if 
; \item 
; \item 
. 
has distinct subset sums if distinct subsets of 
have distinct sums.
so that 
whenever 
has distinct subset sums. 
and every rational 
, there is no polynomial-time algorithm for the following problem.
on 
clauses drawn uniformly at random from the set of formulas on 
simultaneously satisfiable clauses. 
. Suppose that 
. Does it follow that these solutions form a subgroup of 
and 
are 
, then 
which is slice, is it a ribbon knot?
be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and 
be the size of the smallest 
, then 
. 
be an indexed family of filters on sets. Which of the below items are always pairwise equal?
. 
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