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?
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 .
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 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.
Conjecture Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
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 ?
Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.
Conjecture Every arrangement graph of a set of great circles is -colourable.
Conjecture A total coloring of a graph is an assignment of colors to the vertices and the edges of such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph , , equals the minimum number of colors needed in a total coloring of . It is an old conjecture of Behzad that for every graph , the total chromatic number equals the maximum degree of a vertex in , plus one or two. In other words,
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satisfy 
? 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
with 
so that 
has a 
. 
to be the maximum degree, 
to be the size of the largest 
to be the chromatic number of 
for every graph 
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 
is defined to be 
where 
is the number of faces of 
.
so that every convex 4-polytope has fatness at most 
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 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
. The graph 
is defined to be the 
-power of the 
-subdivision of 
.
. Then 
. 
, the 
-th 
, is defined as the sum of all 
by using partitions. 
-connected cubic graph and let 
be a 
is connected. Then 
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
there is an integer 
such that every set of at least 
collinear points or an empty hexagon. 
has a cycle with length at most 
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. 
-colourable. 
is an assignment of colors to the vertices and the edges of 
, equals the minimum number of colors needed in a total coloring of ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
are prime.
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?
so that the power series 
is bounded for all 
. 
denote the graph with vertex set consisting of all lines through the origin in 
and two vertices adjacent in 
? 
there is 
such that, for every large 
such that 
and 
. 
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