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 There is an integer-valued function such that if is any -connected graph and and are any two vertices of , then there exists an induced path with ends and such that is -connected.
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .
Conjecture For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.
Conjecture For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.
Question What is the least integer such that every set of at least points in the plane contains collinear points or a subset of points in general position (no three collinear)?
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 .
Throughout this post, by projective plane we mean the set of all lines through the origin in .
Definition Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.
Problem What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?
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 ?
Problem Is there a minimum integer such that the vertices of any digraph with minimum outdegree can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least ?
Conjecture A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
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is an invertible 
matrix, then there is an 
of ![$ [A A] $](/files/tex/d1e9d82c656535b507686183e640178057fae455.png)
so that 
is nonzero. 
(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
contains a directed path of length 
. 
such that if 
is any 
and 
are any two vertices of 
with ends 
is 
, for 
a large prime, always contain a rainbow 
if each of the color classes is of size of either 
or 
? 
be a set of points in the plane. Two points 
and 
in 
there is an integer 
such that every set of at least 
collinear points or 
(possibly 
) so that the union (as multisets) of any 
contains an additive basis. 
is not greater than 
is a tournament of order 
arc-disjoint transitive subtournaments of order 3. 
such that every set of at least 
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists 
.
such that 
is octahedral. 
and 
such that each set 
is weakly octahedral. Then each 
hold over graphs of bounded tree-width? \item Is 
included in 
over graphs? \item Does 
-uniform hypergraph on 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
-manifold has the homotopy type of the 
, is it diffeomorphic to 
exist a 
such that the vertices of any digraph with minimum outdegree 
, 
does not divide 
where 
is the Legendre symbol. 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
-minor. 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
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