Conjecture Let be a simple -uniform hypergraph, and assume that every set of points is contained in at most edges. Then there exists an -edge-coloring so that any two edges which share vertices have distinct colors.
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?
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 A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
Conjecture It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?
Problem Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .
Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .
There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .
Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .
Conjecture For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.
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,
Conjecture There exists a fixed constant (probably suffices) so that every graft with minimum -cut size at least contains a -join packing of size at least .
Conjecture For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .
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.
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, there is an integer 
so that every strongly 
edge-disjoint Hamilton cycles. 
be a simple 
-
points is contained in at most 
edges. Then there exists an 
-edge-coloring so that any two edges which share 
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? 
. 
is a tournament of order 
arc-disjoint transitive subtournaments of order 3. 
is a directed graph with smallest directed cut of size 
-connected finite graph, is there an assignment of lengths 
to the edges of 
-geodesic cycle is ![$ p \in [0,1] $](/files/tex/1d076f7332523eb59bd7deae19667f83f0a3b6e0.png)
, we let 
denote a subgraph of 
.
so that 
? 
edges whose deletion destroys every odd cycle. 
and a function 
such that if 
, then every saturated 
has cardinality at least 
? 
, is the graph with vertex set 
and two points adjacent if the distance between them is an odd integer. 
? 
Square free? ![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
and 
it holds
if 
if 
; \item 
; \item 
. 
for some constant 
in 
, let the symmetry group of 
ie: isotopy classes of diffeomorphisms of 
of 
is the unlink. Let 
be the subgroup of 
consisting of diffeomorphisms of 
given by restricting the diffeomorphism to the 
is a signed symmetric group -- the wreath product of a symmetric group with 
. 
be a set of points in the plane. Two points 
and 
in 
there is an integer 
is an assignment of colors to the vertices and the edges of 
, equals the minimum number of colors needed in a total coloring of 
plus one or two. In other words, ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
suffices) so that every graft with minimum 
. 
, what is the smallest integer 
such that 
? 
. 
of 
lines determined by 
has a 
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