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 If is a cubic graph not containing a triangle, then it is possible to color the edges of by five colors, so that the complement of every color class is a bipartite graph.
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 ?
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?
For a finite (additive) abelian group , the Davenport constant of , denoted , is the smallest integer so that every sequence of elements of with length has a nontrivial subsequence which sums to zero.
Problem Let be positve integer Does there exists an integer such that every -strong tournament admits a partition of its vertex set such that the subtournament induced by is a non-trivial -strong for all .
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 .
An oriented colouring of an oriented graph is assignment of colours to the vertices such that no two arcs receive ordered pairs of colours and . It is equivalent to a homomorphism of the digraph onto some tournament of order .
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?
Problem Given two codes , their Tensor Product is the code that consists of the matrices whose rows are codewords of and whose columns are codewords of . The product is said to be robust if whenever a matrix is far from , the rows (columns) of are far from (, respectively).
The problem is to give a characterization of the pairs whose tensor product is robust.
Let be a hypergraph. A strongly maximal matching is a matching so that for every matching . A strongly minimal cover is a (vertex) cover so that for every cover .
Conjecture If is a (possibly infinite) hypergraph in which all edges have size for some integer , then has a strongly maximal matching and a strongly minimal cover.
Conjecture For every graph without a bridge, there is a flow .
Conjecture There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.
Conjecture Denote by the number of non-Hamiltonian 3-regular graphs of size , and similarly denote by the number of non-Hamiltonian 3-regular 1-connected graphs of size .
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, 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 
.
is a cubic graph not containing a triangle, then it is possible to color the edges of 
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 
with 
so that 
has a 
. 
which is slice, is it a ribbon knot?
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 
pairwise visible points. 
, is the smallest integer 
so that every sequence of elements of 
has a nontrivial subsequence which sums to zero.
be positve integer Does there exists an integer 
such that every 
admits a partition 
of its vertex set such that the subtournament induced by 
is a non-trivial 
-strong for all 
. 
, 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 complete bipartite subgraph with bipartition 
so that 
? 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
be the unique integer (with respect to a fixed 
) such that
is a prime iff ![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
arc-disjoint directed cycles. 
contains 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
, their Tensor Product 
is the code that consists of the matrices whose rows are codewords of 
and whose columns are codewords of 
. The product 
is far from 
be a 
so that 
for every matching 
. A strongly minimal cover is a (vertex) cover 
so that 
for every cover 
. 
for some integer 
.
so that antipodal points of 
receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero. 
edges whose deletion destroys every odd cycle. 
is the only 
the number of non-Hamiltonian 3-regular graphs of size 
, and similarly denote by 
the number of non-Hamiltonian 3-regular 1-connected graphs of size 
? 
, and let 
be the edge set of a forest chosen uniformly at random from all forests of ![\[ {\mathbb P}(e \in F \mid f \in F}) \le {\mathbb P}(e \in F) \]](/files/tex/1fe7ac9c579238670e3aa16ae401e8194e1c3da3.png)
be an integer and let 
denote the least integer 
. 
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