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 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 .
Conjecture There exists a fixed constant so that every abelian group has a subset with so that the Cayley graph has no clique or independent set of size .
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 .
Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
\item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.
See details, exact definitions, and attempted proofs here.
Conjecture Let be a field of characteristic zero. A collection of polynomials in variables defines an automorphism of if and only if the Jacobian matrix is a nonzero constant.
Setup Fix a tree and for every vertex a non-negative integer which we think of as the amount of gold at .
2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex of the tree, takes the gold at this vertex, and then deletes . The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.
Note that the above is a generalization of monotone Galois connections (with and replaced with suprema and infima).
Then we have the following diagram:
What is at the node "other" in the diagram is unknown.
Conjecture "Other" is .
Question What repeated applying of and to "other" leads to? Particularly, does repeated applying and/or to the node "other" lead to finite or infinite sets?
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is 
-degenerate if every subgraph of 
.
, the union of any 
-degenerate graph. 
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)
be a set of 
positive integers and set ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
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 
. Does the fragment 
have the finite model property? 
(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
to be the maximum degree, 
to be the size of the largest 
to be the chromatic number of 
for every graph 
with 
so that the 
has no clique or independent set of size 
. 
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists 
such that 
-manifold has the homotopy type of the 
, is it diffeomorphic to 
. 
-regular graph, where 
, then 
of polynomials in variables 
defines an automorphism of 
if and only if the Jacobian matrix is a nonzero constant. 
and for every vertex 
a non-negative integer 
which we think of as the amount of gold at 
. 
and 
are 
, then 
be the flow polynomial of a graph 
equals the number of 
for every 2-edge-connected graph 
has distinct subset sums if distinct subsets of 
have distinct sums.
whenever 
has distinct subset sums. 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
. The graph 
is defined to be the 
-power of the 
.
. Then 
. 
with 
so that 
has a 
. 
does there exists a 
such that every graph with average degree smaller than 
such that its fundamental group has an element of finite order?
of co-prime positive integers 
for 
. 
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