If is a finite set of points which is 2-colored, an empty triangle is a set with so that the convex hull of is disjoint from . We say that is monochromatic if all points in are the same color.
Conjecture There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.
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 Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.
Conjecture Let be an integer. For every integer , there exists an integer such that for every digraph , either has a pairwise-disjoint directed cycles of length at least , or there exists a set of at most vertices such that has no directed cycles of length at least .
Given integers , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).
Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).
Let be the smallest integer such that the graph is rearrangeable.
Conjecture Let be the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space
This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .
The crossing number of is the minimum number of crossings in all drawings of in the plane.
The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.
Problem has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .
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.
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for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
is a tournament of order 
, then it contains 
arc-disjoint transitive subtournaments of order 3. 
and a function 
such that if 
, then every saturated 
-Sperner system 
has cardinality at least 
? 
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
. We say that 
with the following property. If 
monochromatic empty triangles. 
of 
lines determined by 
-connected cubic graph on 
vertices admit a partition into 
? 
can be represented as the sum of an odd prime number and an odd 
, is the graph with vertex set 
and two points adjacent if the distance between them is an odd integer. 
? 
be a 
and 
are 
, then 
concentrated on a single value? 
be a circuit in a bridgeless cubic graph 
. Then there is a five cycle double cover of 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, either 
, or there exists a set 
vertices such that 
has no directed cycles of length at least 
, the 2-stage Shuffle-Exchange graph/network, denoted 
, is the simple 
of linearly labeled parts 
and 
, where 
, such that vertices 
and 
are adjacent if and only if 
(see Fig.1).
, the 
-stage Shuffle-Exchange graph/network, denoted 
, is the proper (i.e., respecting all the orders) concatenation of 
identical copies of 
be the smallest integer 
such that the graph 
. 
be the space of 
Diffeomorphisms on the connected , compact and boundaryles manifold M and 
the space of 
(
) such that 
exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space 
which cannot be expressed as a union of 
triangle free graphs? 
are matroids on 
and 
for every partition 
of 
with 
which is independent in every 
. 
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
in the hypercube. 
of 
-dimensional (hyper)cube 
is the graph whose vertices are all binary sequences of length 
for principal funcoid 
and a set 
of funcoids of appropriate sources and destinations. 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
satisfy 
. Then the elements of 
and 
may be ordered 
and 
so that the sums 
are pairwise distinct. 
people admits a solution is 
. 
. The graph 
is defined to be the 
-power of the 
.
. Then 
. 
is arbitrarily large? 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
has the homotopy-type of a product space 
where 
is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of 
, for 
a large prime, always contain a rainbow 
if each of the color classes is of size of either 
or 
? 
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. 
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