Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
Conjecture Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
Conjecture Let be the disjoint union of the graphs and and let be a surface. Is it true that every optimal drawing of on has the property that and are disjoint?
Problem Find a constant such that for any there is a sequence of tautologies of depth that have polynomial (or quasi-polynomial) size proofs in depth Frege system but requires exponential size proofs.
Problem Let and be two -uniform hypergraph on the same vertex set . Does there always exist a partition of into classes such that for both , at least hyperedges of meet each of the classes ?
Conjecture Let be an -uniform-partite hypergraph. If is the maximum number of pairwise disjoint edges in , and is the size of the smallest set of vertices which meets every edge, then .
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 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 .
A strong edge-colouring of a graph is a edge-colouring in which every colour class is an induced matching; that is, any two vertices belonging to distinct edges with the same colour are not adjacent. The strong chromatic index is the minimum number of colours in a strong edge-colouring of .
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does there exists a 
such that every graph with average degree smaller than 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
on 
) 
(The join operation is taken on the lattice of filters with reversed order.) 
, then 
contains at least 
cycles of length 
. 
if 
is odd and 
if 
. Assume we start with some number 
of the current number. Prove that no matter what the initial number is we eventually reach 
. 
such that every 
-minor-free graph has at most 
cliques? 
and 
and let 
be a surface. Is it true that every optimal drawing of 
such that for any 
Frege system 
but requires exponential size 
proofs. 
such that 
or prove that such integers do not exist. 
? 
be a digraph. If 
, then 
be an abelian group. Is the 
and 
be two 
-uniform hypergraph on the same vertex set 
. Does there always exist a partition of 
such that for both 
, at least 
hyperedges of 
meet each of the classes 
be an 
is the maximum number of pairwise disjoint edges in 
is the size of the smallest set of vertices which meets every edge, then 
. 
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 
on 
from the edge set of 
, for all 
, and 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
, every 
vertices has a Hamilton cycle. 
which is slice, is it a ribbon knot?
is a tournament of order 
arc-disjoint transitive subtournaments of order 3. 
-size circuits compute the function 
defined inductively by 
, 
, 
, and 
? 
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 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
contains 
is the minimum number of colours in a strong edge-colouring of 
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