Let be a non-empty finite set. Given a partition of , the stabilizer of , denoted , is the group formed by all permutations of preserving each block of .
Problem () Find a sufficient condition for a sequence of partitions of to be complete, i.e. such that the product of their stabilizers is equal to the whole symmetric group on . In particular, what about completeness of the sequence , given a partition of and a permutation of ?
Conjecture (Beneš) Let be a uniform partition of and be a permutation of such that . Suppose that the set is transitive, for some integer . Then
Conjecture Let and . Then for any neighborhood there is such that is periodic point of
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for
Conjecture Let be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists such that is an element of at least half the members of .
Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .
Conjecture If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
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.
Conjecture Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .
Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.
Question Can either of the following be expressed in fixed-point logic plus counting: \item Given a graph, does it have a perfect matching, i.e., a set of edges such that every vertex is incident to exactly one edge from ? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
Conjecture Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.
In particular, this implies:
Conjecture Twin Prime Conjecture: There are an infinite number of twin primes.
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 a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?
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.
Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is
Problem What is the best upper bound on circular choosability for planar graphs?
Conjecture Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .
Conjecture For every fixed graph , there exists a constant , so that every graph without an induced subgraph isomorphic to contains either a clique or an independent set of size .
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be a non-empty finite set. Given a partition 
of 
, is the group formed by all permutations of 
.
) Find a sufficient condition for a sequence of partitions 
of 
is equal to the whole symmetric group 
on 
, given a partition 
of 
be a uniform partition of 
be a permutation of 
. Suppose that the set 
is transitive, for some integer 
. Then 
and 
. Then for any neighborhood 
there is 
such that 
is periodic point of 
vector fields . In the case of diffeos this was proved by Charles Pugh for 
. In the case of Flows this has been solved by Sushei Hayahshy for 
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists 
such that 
be a set, 
be the set of filters on 
be the set of principal filters on 
be an index set. Consider the filtrator 
.
is a completary multifuncoid of the form 
, then 
is a completary multifuncoid of the form 
. 
(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
is called double-critical, if removing any pair of adjacent vertexes lowers the 
is the only 
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. 
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
, where 
is some ordinal. This 
as 
.
concentrated on a single value? 
of edges such that every vertex is incident to exactly one edge from 
for every filter objects 
? 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
or 
which divides both 
(see 
(see 
grid. The first player (if any) to occupy a set of 
has distinct subset sums if distinct subsets of 
have distinct sums.
whenever 
has distinct subset sums. 
. In other words, simplify this formula. 
be a graph. If 
-colouring of 
to 
such that 
for each edge 
. Given a list assignment 
of 
a set of non-negative integers, an 
such that 
for every 
. A list assignment 
-
and 
for each vertex 
. Given such a list assignment 
, the graph 
for reloid 
for every funcoid 
(see below mentioned online article).
points in the plane in general position contains a subset of ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
such that for any pair 
there are 
edge-disjoint paths from 
to 
edge-disjoint trees, each of which contains 
. 
, there exists a constant 
, so that every graph 
. 
.
are matroids on 
for every partition 
of 
with 
which is independent in every 
. 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
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