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 Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.
Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.
Problem Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?
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 ?
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denote the 
.
satisfies 
. 
which cannot be expressed as a union of 
triangle free graphs? 
, it is impossible to cover a square of side greater than 
with 
unit squares. 
such that its fundamental group has an element of finite order?
-diregular if every vertex has indegree and outdegree at least 
, every 
-diregular oriented graph on at most 
vertices has a Hamilton cycle. 
has a proper 
-
so that all pairwise distances between points in 
are rational? 
is an invertible 
matrix, then there is an 
of ![$ [A A] $](/files/tex/d1e9d82c656535b507686183e640178057fae455.png)
so that 
is nonzero. 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
there is 
such that, for every large 
such that 
and 
. 
denote the pebbling number of a graph 
. 
, then 
cycles of length 
. 
-connected planar graph, then 
has a Hamilton cycle. 
be natural numbers with 
. It follows from the pigeon-hole principle that there exist distinct subsets 
with 
. Is it possible to find such a pair 
in polynomial time? 
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 
paths. 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
is defined to be the mimic number of any positive integer 
. 
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