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 .
Conjecture For all positive integers and , there exists an integer such that every graph of average degree at least contains a subgraph of average degree at least and girth greater than .
Problem Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .
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.
Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .
Conjecture If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Conjecture There exists a fixed constant so that every abelian group has a subset with so that the Cayley graph has no clique or independent set of size .
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.
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? 
people admits a solution is 
. 
, 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 
. 
contains a directed path of length 
. 
we have 
. Then 
is graphic. 
and 
such that every graph of average degree at least 
, there is an integer 
so that every strongly 
-
(mod 
) for every vertex 
. 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
be a graph, 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
? 
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
be a set of points in the plane. Two points 
in 
there is an integer 
such that every set of at least 
collinear points or 
is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. 
is a 2-transition system for 
has a compaible decomposition. 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
is a completary multifuncoid of the form 
, then 
is a completary multifuncoid of the form 
. 
with 
so that the 
has no clique or independent set of size 
. 
denote the 
is called edge-antipodal if 
whenever 
are antipodal edges. 
and 
which are joined by a monochromatic path. 
of 
the following inequality holds 
denote the 
diagonal 
exists. 
is for every sets 
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 
monochromatic empty triangles. 
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