Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
Question What is the least integer such that every set of at least points in the plane contains collinear points or a subset of points in general position (no three collinear)?
Problem Is there a minimum integer such that the vertices of any digraph with minimum outdegree can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least ?
Problem Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?
Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .
A permutation is called a Roller Coaster permutation if . Let be the set of all Roller Coaster permutations in .
Conjecture For ,
\item If , then . \item If , then with .
Conjecture (Odd Sum conjecture) Given ,
\item If , then is odd for . \item If , then for all .
Conjecture Let be a Cantor set embedded in . Is there a self-homeomorphism of for every greater than so that moves every point by less than and does not intersect ? Such an embedded Cantor set for which no such exists (for some ) is called "sticky". For what dimensions do sticky Cantor sets exist?
Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
Conjecture For every fixed graph , there exists a constant , so that every graph without an induced subgraph isomorphic to contains either a clique or an independent set of size .
Conjecture Suppose that is a -edge-critical graph. Suppose that for each edge of , there is a list of colors. Then is -edge-colorable unless all lists are equal to each other.
Conjecture Let be an eulerian graph of minimum degree , and let be an eulerian tour of . Then admits a decomposition into cycles none of which contains two consecutive edges of .
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 .
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
An alternating walk in a digraph is a walk so that the vertex is either the head of both and or the tail of both and for every . A digraph is universal if for every pair of edges , there is an alternating walk containing both and
Question Does there exist a locally finite highly arc transitive digraph which is universal?
An -factor in a graph is a set of vertex-disjoint copies of covering all vertices of .
Problem Let be a fixed positive real number and a fixed graph. Is it NP-hard to determine whether a graph on vertices and minimum degree contains and -factor?
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 Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
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 .
Conjecture For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .
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contains a directed path of length 
. 
? 
is the total graph of a multigraph, then 
. 
such that every set of at least 
collinear points or a subset of 
, whether there exists a homomorphism from 
for some constant 
? 
such that the vertices of any digraph with minimum outdegree 
can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least 
be an indexed family of sets.
for 
.
on the lattice 
of all finite unions of products.
a bijection from hyperfuncoids 
defining the inverse bijection? If not, characterize the image of the function 
of complements of elements of the set 
? 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
where 
is the number of vertices. What is the distribution of 
for 
? 
of co-prime positive integers 
for 
. 
is arbitrarily large? 
-connected cubic planar graph can be decomposed into two Hamilton cycles. 
, then 
cycles of length 
. 
-manifold has the homotopy type of the 
, is it diffeomorphic to 
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
denote the set of all permutations of ![$ [n]=\set{1,2,\ldots,n} $](/files/tex/470bd09059264370b76b4da90b77dc370c7e0e7c.png)
. Let 
and 
denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in 
. Let 
denote the set of subsequences of 
denote 
.
is called a Roller Coaster permutation if 
. Let 
be the set of all Roller Coaster permutations in 
,
, then 
. \item If 
, then 
with 
. 
,
is odd for 
. \item If 
for all 
be a 
. Is there a self-homeomorphism 
of 
so that 
does not intersect 
. Suppose X is an m-by-n integer matrix 
. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
, so that every graph 
. 
-edge-critical graph. Suppose that for each edge 
of 
of 
, the 
, is defined as the sum of all 
by using partitions. 
be an eulerian tour of 
or 
which divides both 
(see 
(see 
vertices, then 
. 
so that the vertex 
is either the head of both 
and 
or the tail of both 
. A digraph is universal if for every pair of edges 
, there is an alternating walk containing both 
contains and 
and 
. It is equivalent to a homomorphism of the digraph onto some tournament of order 
are independent random variables with ![$ \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)
if 
if 
. Assume we start with some number 
. 
a countable end of 
of pairwise disjoint 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to 
always rational? 
satisfy 
. Then the elements of 
may be ordered 
and 
so that the sums 
are pairwise distinct. 
-minor-free graph has at most 
cliques? 
of 
lines determined by 
-graph is an 
for every 
with odd size.
for every 
is for every sets 
and 
be an integer and let 
. Then either 
minor for some 
or 
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