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 .
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 ?
Conjecture Suppose runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, there is a time at which that runner is distance at least (along the track) away from every other runner.
Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.
Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.
Conjecture If is the maximal degree of a graph , then is strongly -colorable.
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.
Question Can either of the following be expressed in fixed-point logic plus counting: \item Given a graph, does it have a perfect matching, i.e., a set of edges such that every vertex is incident to exactly one edge from ? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?
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.
A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.
Problem Is it true that for every , all but finitely many -regular graphs have friendly partitions?
Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.
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the following inequality holds 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
let 
denote the minimal number 
such that there is a rainbow 
in every equinumerous 
-coloring of 
for every 
, 
. 
is for every sets 
and 
:
, let 
be the statement that given any exact 
-coloring of the edges of a complete countably infinite graph (that is, a coloring with 
, 
, or 
. 
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 
that is a minor-minimal obstruction for two non-orientable surfaces? 
edge-disjoint triangles, then there is a set of 
edges whose deletion destroys every triangle. 
(along the track) away from every other runner. 
is the maximal degree of a graph 
-colorable. 
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 
. 
-uniform hypergraph on 
vertices which contains no complete 
hyperedges. 
vertices which contains no complete 
hyperedges. 
of edges such that every vertex is incident to exactly one edge from 
satisfy 
. Then the elements of 
and 
may be ordered 
and 
so that the sums 
are pairwise distinct. 
has distinct subset sums if distinct subsets of 
have distinct sums.
whenever 
has distinct subset sums. 
contains a complete bipartite subgraph with bipartition 
so that 
? 
contains every oriented tree of order 
is a prime iff 
and 
in 
there is an integer 
collinear points or 
on an even number of vertices can be decomposed into 
directed paths. 
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