Conjecture Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .
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 An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the vertices' trees, the star with vertices has the most endomorphisms, while the path with vertices has the least endomorphisms.
Conjecture Can the approximation ratio be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than -hardness?
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?
Question \item Does hold over graphs of bounded tree-width? \item Is included in over graphs? \item Does have a 0-1 law? \item Are properties of Hanf-local? \item Is there a logic (with an effective syntax) that captures ?
Conjecture Let and . Then for any neighborhood there is such that is periodic point of
There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
Conjecture If is a cubic graph not containing a triangle, then it is possible to color the edges of by five colors, so that the complement of every color class is a bipartite graph.
Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.
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, where 
is some ordinal. This 
as 
.
be a prime natural number. Find all primes 
, such that 
.
be a matroid of rank 
, and for 
let 
be the number of closed sets of rank 
.
is unimodal. 
\item 
is a fixed recursive set of integers.
and a closed formula 
in this language.
of tree-width at most 
concentrated on a single value? 
. In other words, simplify this formula. 
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
, the number of even latin squares of order 
such that 
meets 
in at most one vertex, 
. 
. 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
hold over graphs of bounded tree-width? \item Is 
included in 
over graphs? \item Does 
and 
. Then for any neighborhood 
there is 
such that 
vector fields . In the case of diffeos this was proved by Charles Pugh for 
. In the case of Flows this has been solved by Sushei Hayahshy for 
can be represented as the sum of an odd prime number and an odd 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 
around the north and south poles. 
on an even number of vertices can be decomposed into 
directed paths. 
and 
has distinct subset sums if distinct subsets of 
have distinct sums.
so that 
whenever 
has distinct subset sums. 
are simple finite graphs, then 
. 
is the 
.
to the edges of 
-geodesic cycle is 
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