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 \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.
Conjecture If in a bridgeless cubic graph the cycles of any -factor are odd, then , where denotes the oddness of the graph , that is, the minimum number of odd cycles in a -factor of .
Problem The valency-variety of a graph is the number of different degrees in . Is the chromatic number of any graph with at least two vertices greater than
Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a deterministic Turing machine in polynomial time? That is, does P = BPP?
Conjecture For every positive integer , there exists an integer so that every polytope of dimension has a -dimensional face which is either a simplex or is combinatorially isomorphic to a -dimensional cube.
Problem Let and be two -uniform hypergraph on the same vertex set . Does there always exist a partition of into classes such that for both , at least hyperedges of meet each of the classes ?
Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?
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?
The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Conjecture Define a array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array , the sequence is: -> -> -> -> -> -> -> -> -> -> -> , and we now have a fixed point (loop of one array).
The process always results in a loop of 1, 2, or 3 arrays.
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is a 4-edge-connected locally finite graph, then its 
of a locally finite graph 
edge-disjoint triangles, then there is a set of 
edges whose deletion destroys every triangle. 
is for every sets 
and 
:
? 
-factor are odd, then 
, where 
denotes the oddness of the graph 
there is an integer 
such that every set of at least 
collinear points or an empty hexagon. 
of a graph 
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
with 
so that 
has a 
. 
-neighbour bootstrap process in the hypercube. 
so that every polytope of dimension 
has a 
and 
be two 
-uniform hypergraph on the same vertex set 
. Does there always exist a partition of 
such that for both 
, at least 
hyperedges of 
meet each of the classes 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
always rational? 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
rectangular 
inside an 
? 
array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array ![$ [1; 1] $](/files/tex/73f8649de444361674c157a2fe98e0c5783f1c46.png)
, the sequence is: ![$ [1; 2] $](/files/tex/83c3d9d7589f716ed6b0f05d26d36dabe8ba47aa.png)
-> ![$ [1, 2; 1, 1] $](/files/tex/a6a696aec4e84df6bc046cf6d30a4df80e156a14.png)
-> ![$ [1, 2; 3, 1] $](/files/tex/98b2f3e4134422c8a286f0326fc2f57ca9be2ab7.png)
-> ![$ [1, 2, 3; 2, 1, 1] $](/files/tex/19a9d24510f4551fa45b950aed32efed0636b355.png)
-> ![$ [1, 2, 3; 3, 2, 1] $](/files/tex/b00d573110ba8c2472820409103dd4cbd7bff7cc.png)
-> ![$ [1, 2, 3; 2, 2, 2] $](/files/tex/2aae90b65ae3f116402a1c0143127f80d86acdf0.png)
-> ![$ [1, 2, 3; 1, 4, 1] $](/files/tex/d614399c4078a70fbffb24eb99816a0974423175.png)
-> ![$ [1, 2, 3, 4; 3, 1, 1, 1] $](/files/tex/bf4dd82eaf95bd71a88cb78d3367efa2c7e4c942.png)
-> ![$ [1, 2, 3, 4; 4, 1, 2, 1] $](/files/tex/a4ef5900a9dde9b32311afe7dee2b143f42c405f.png)
-> ![$ [1, 2, 3, 4; 3, 2, 1, 2] $](/files/tex/ff367376ed7bb74fc6207273d7157062b83a1be6.png)
-> ![$ [1, 2, 3, 4; 2, 3, 2, 1] $](/files/tex/81ba2b3608ddba2fc95d32b74b70279f3f5adc5b.png)
, and we now have a fixed point (loop of one array).
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