Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .
Question Does the Blatter-Specker Theorem hold for ternary relations.
If is a finite set of points which is 2-colored, an empty triangle is a set with so that the convex hull of is disjoint from . We say that is monochromatic if all points in are the same color.
Conjecture There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.
Conjecture Let is a family of multifuncoids such that each is of the form where is an index set for every and is a set for every . Let every for some multifuncoid of the form regarding the filtrator . Let is a graph-composition of (regarding some partition and external set ). Then there exist a multifuncoid of the form such that regarding the filtrator .
Conjecture Let be the open unit disk in the complex plane and let be open sets such that . Suppose there are injective holomorphic functions such that for the differentials we have on any intersection . Then those differentials glue together to a meromorphic 1-form on .
Conjecture Let be a Cantor set embedded in . Is there a self-homeomorphism of for every greater than so that moves every point by less than and does not intersect ? Such an embedded Cantor set for which no such exists (for some ) is called "sticky". For what dimensions do sticky Cantor sets exist?
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 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 A total coloring of a graph is an assignment of colors to the vertices and the edges of such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph , , equals the minimum number of colors needed in a total coloring of . It is an old conjecture of Behzad that for every graph , the total chromatic number equals the maximum degree of a vertex in , plus one or two. In other words,
Conjecture The following statements are equivalent for every endofuncoid and a set : \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .
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?
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such that its fundamental group has an element of finite order?
be a class of finite relational structures. We denote by 
the number of structures in 
. For any class 
, the function 
. 
the following inequality holds 
-degenerate graph (a forest) and a 
-degenerate graph is 
-colourable. 
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
is disjoint from 
. We say that 
with the following property. If 
points in general position which is 2-colored, then it has 
monochromatic empty triangles. 
are invertible 
matrices with entries in 
for a prime 
, then there is a 
submatrix 
of ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
is a family of multifuncoids such that each 
is of the form 
where 
is an index set for every 
and 
is a set for every 
. Let every 
for some multifuncoid 
of the form 
regarding the filtrator 
. Let 
is a graph-composition of 
and external set 
). Then there exist a multifuncoid 
of the form 
such that 
regarding the filtrator 
. 
there is an integer 
such that every set of at least 
collinear points or an empty hexagon. 
be the open unit disk in the complex plane and let 
be open sets such that 
. Suppose there are injective holomorphic functions 
such that for the differentials we have 
on any intersection 
. Then those differentials glue together to a meromorphic 1-form on 
. Is there a self-homeomorphism 
of 
greater than 
so that 
does not intersect 
be a set of points in the plane. Two points 
and 
in 
there is an integer 
pairwise visible points. 
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.) 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
is an assignment of colors to the vertices and the edges of 
, equals the minimum number of colors needed in a total coloring of 
plus one or two. In other words, ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
for reloid 
for every funcoid 
(see below mentioned online article).
vertices. 
. 
be a 
.
and a set 
:
there exists a totally ordered set 
such that 
, 
, and for every partion 
of 
into two sets 
, 
such that 
, we have ![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
. 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
has chromatic number at most 
. 
contains a directed cycle of length at least 
. 
be a matroid of rank 
, and for 
let 
be the number of closed sets of rank 
is unimodal. 
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