Conjecture For all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order .
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 .
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .
A permutation is called a Roller Coaster permutation if . Let be the set of all Roller Coaster permutations in .
Conjecture For ,
\item If , then . \item If , then with .
Conjecture (Odd Sum conjecture) Given ,
\item If , then is odd for . \item If , then for all .
Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?
Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.
Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?
2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?
3. Finally, what could one say about higher dimensional analogs of this question?
Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n
Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.
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 ?
Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .
Conjecture [2] Let be a graph with list chromatic number and . Then
Conjecture There is an integer-valued function such that if is any -connected graph and and are any two vertices of , then there exists an induced path with ends and such that is -connected.
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
Problem Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?
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be the flow polynomial of a graph 
. So for every positive integer 
, the value 
equals the number of 
for every 2-edge-connected graph 
such that every digraph of minimum outdegree at least 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
of colours to the vertices such that no two arcs receive ordered pairs of colours 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
are independent random variables with ![$ \mathbb{E}[X_i] \leq \mu $](/files/tex/e0268221532981debea25e9446c8ee6f112e1881.png)
, then ![$$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$](/files/tex/03dc1130142ee6fefcc33888e2fb6137211bf327.png)
with 
so that 
has a 
. 
contains every oriented tree of order 
, whether there exists a homomorphism from 
for some constant 
, let 
be the smallest integer 
such that the symmetric group 
on the set of all words of length 
over a 
(
times), where 
is the shuffle permutation defined by 
, and 
is the exchange group consisting of all permutations in 
letters in the words. 
, for all 
denote the set of all permutations of ![$ [n]=\set{1,2,\ldots,n} $](/files/tex/470bd09059264370b76b4da90b77dc370c7e0e7c.png)
. Let 
and 
denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in 
. Let 
denote the set of subsequences of 
denote 
.
is called a Roller Coaster permutation if 
. Let 
be the set of all Roller Coaster permutations in 
,
, then 
. \item If 
, then 
with 
. 
,
is odd for 
. \item If 
for all 
are simple finite graphs, then 
. 
is the 
concentrated on a single value? 
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 
? 
let 
be the maximum number of vertices of 
, where the minimum is taken over all list assignments 
for all 
. 
and 
. Then ![\[\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.\]](/files/tex/47be18e956355dd433b88b66eabf01a9e3ed5f61.png)
. 
and 
are any two vertices of 
with ends 
is 
be an integer and let 
denote the least integer 
, that every sufficiently large 
vertices whose deletion results in a 
-connected graph with a Hamilton cycle has a second Hamilton cycle. 
be a 
-dimensional smooth submanifold of 
, 
. By the Jordan-Brouwer separation theorem, 
? ie: is 
. The graph 
is defined to be the 
-power of the 
.
. Then 
. 
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