We say that a set is -universal if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.
Question Does there exist an -universal set of size ?
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 If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.
Conjecture Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.
Conjecture For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .
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?
The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)
Problem Does there exist a polynomial time algorithm which takes as input a graph and for every vertex a subset of , and decides if there exists a partition of into so that only if and so that are independent, is a clique, and there are no edges between and ?
Conjecture For every , the sequence in consisting of copes of and copies of has the fewest number of distinct subsequence sums over all zero-free sequences from of length .
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?
Conjecture It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?
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 .
Conjecture Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.
Conjecture Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in -outerplanar graphs or tree-width graphs?
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
Conjecture There is a function such that for every graph ,
Conjecture For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .
Given a finite family of graphs and an integer , the Turán number of is the largest integer such that there exists a graph on vertices with edges which contains no member of as a subgraph.
Conjecture For every finite family of graphs there exists an such that .
Problem The valency-variety of a graph is the number of different degrees in . Is the chromatic number of any graph with at least two vertices greater than
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. The representation function 
for 
is given by the rule 
. We call 
is never 
.
is 
-universal if every 
, and all edges are (non-intersecting) straight line segments. 
? 
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 
. 
. 
, and let ![\[ {\mathbb P}(e \in F \mid f \in F}) \le {\mathbb P}(e \in F) \]](/files/tex/1fe7ac9c579238670e3aa16ae401e8194e1c3da3.png)
, then there is a 
-vertex colouring of 
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
-colored countably infinite complete subgraph. Then 
, 
, or 
.
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
a subset 
of 
, and decides if there exists a partition of 
into 
so that 
only if 
and so that 
are independent, 
is a clique, and there are no edges between 
and 
? 
is for every sets 
and 
, the sequence in 
consisting of 
copes of 
and 
copies of 
has the fewest number of distinct subsequence sums over all zero-free sequences from 
. 
grid. The first player (if any) to occupy a set of 
-outerplanar embedding for which 
denote the 
diagonal 
exists. 
such that 
-connected cubic graph on 
vertices admit a partition into 
be a circuit in a bridgeless cubic graph 
such that the list chromatic number of any bipartite graph 
is at most 
.
of 
such that edges with the same colour do not cross with respect to 
for some edges 
, then 
and 
receive distinct colours. 
is the minimum integer 
be the graph obtained by subdividing each edge of 
such that for every graph 
is 
. In other words, simplify this formula. 
, let 
be the statement that given any exact 
-colored countably infinite complete subgraph. Then 
, 
, or 
. 
defined inductively by 
, 
, 
, and 
? 
of graphs and an integer 
of 
such that 
.
concentrated on a single value? 
of a graph 
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