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 .
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.
Problem Does the following equality hold for every graph ?
The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number, we minimize the number of pairs of edges that cross.
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.
Problem (2) Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .
Conjecture Let be a simple -uniform hypergraph, and assume that every set of points is contained in at most edges. Then there exists an -edge-coloring so that any two edges which share vertices have distinct colors.
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
Problem has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .
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 .
Suppose is a finite group, and is a positive integer dividing . Suppose that has exactly solutions to . Does it follow that these solutions form a subgroup of ?
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 .
Conjecture Let be a sequence of points in with the property that for every , the points are distinct, lie on a unique sphere, and further, is the center of this sphere. If this sequence is periodic, must its period be ?
The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
Conjecture If two graphs on vertices have the same deck, then they are isomorphic.
Conjecture For every , there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .
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![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
, there exist positive integers 
, 
, 
such that 
.
be a graph, 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint ![\[ \text{pair-cr}(G) = \text{cr}(G) \]](/files/tex/8cece1e00bb0e9fc122e0a5cad0dab2681cf33a4.png)
of a graph 
, we minimize the number of pairs of edges that cross. 
, 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 
such that 
or 
which divides both 
(see 
(see 
with 
so that 
has a 
. 
be a simple 
-
points is contained in at most 
edges. Then there exists an 
-edge-coloring so that any two edges which share 
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
, 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)
, what is the smallest integer 
such that 
? 
has the homotopy-type of a product space 
where 
is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of 
. Suppose that 
. Does it follow that these solutions form a subgroup of 
has chromatic number at most 
. 
be a digraph. If 
, then 
. 
, the sequence in 
consisting of 
copes of 
and 
copies of 
has the fewest number of distinct subsequence sums over all zero-free sequences from 
. 
be abelian groups and let 
and 
satisfy 
and 
. If there is a 
to 
, then every graph with a B-flow has a B'-flow. 
be a sequence of points in 
with the property that for every 
, the points 
are distinct, lie on a unique sphere, and further, 
is the center of this sphere. If this sequence is periodic, must its period be 
? 
? 
of co-prime positive integers 
for 
. 
vertices have the same deck, then they are isomorphic. 
are independent random variables with ![$ \mathbb{E}[X_i] \leq \mu $](/files/tex/e0268221532981debea25e9446c8ee6f112e1881.png)
, then ![$$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$](/files/tex/03dc1130142ee6fefcc33888e2fb6137211bf327.png)
has a complete graph on 
vertices as a minor. 
such that if 
set which is the union of 
and every positive integer 
so that every simple 
has a 
. 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
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