Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
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 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.
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?
Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
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 .
The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:
\item \item
where is a fixed recursive set of integers.
Let us fix and a closed formula in this language.
Conjecture Is it true that the validity of for a graph of tree-width at most can be tested in polynomial time in the size of ?
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
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Square free? 
contains every oriented tree of order 
as a subdigraph. 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
be a perfect graph on 
vertices. Is it true that either 
contains a complete bipartite subgraph with bipartition 
so that 
? 
and graph 
. 
contains every 2-regular graph on 
vertices as a minor. 
of 
-dimensional (hyper)cube 
is the graph whose vertices are all binary sequences of length 
satisfy 
. Then the elements of 
and 
may be ordered 
and 
so that the sums 
are pairwise distinct. 
Frege system 
but requires exponential size 
proofs. 
is the minimum number 
for which there exists a polytope 
with 
with 
.
? 
denote the graph with vertex set consisting of all lines through the origin in 
and two vertices adjacent in 
? 
is a 
for all primes 
, where 
is prime. 
contains 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
be a digraph all of whose strong components are nontrivial. Then 
. 
there is 
such that, for every large 
and 
. 
be an integer and let 
denote the least integer 
. 
is the minimum number of colours in a strong edge-colouring of 
\item 
in this language.
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
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