Conjecture Define a array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array , the sequence is: -> -> -> -> -> -> -> -> -> -> -> , and we now have a fixed point (loop of one array).
The process always results in a loop of 1, 2, or 3 arrays.
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)?
Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
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 .
Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?
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 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.
Conjecture If is a finite field with at least 4 elements and is an invertible matrix with entries in , then there are column vectors which have no coordinates equal to zero such that .
Conjecture For every fixed and fixed colouring of with colours, there exists such that every colouring of the edges of contains either vertices whose edges are coloured according to or vertices whose edges are coloured with at most colours.
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 .
Conjecture Given any complex numbers which are linearly independent over the rational numbers , then the extension field has transcendence degree of at least over .
Conjecture Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.
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, then 
contains at least 
cycles of length 
. 
array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array ![$ [1; 1] $](/files/tex/73f8649de444361674c157a2fe98e0c5783f1c46.png)
, the sequence is: ![$ [1; 2] $](/files/tex/83c3d9d7589f716ed6b0f05d26d36dabe8ba47aa.png)
-> ![$ [1, 2; 1, 1] $](/files/tex/a6a696aec4e84df6bc046cf6d30a4df80e156a14.png)
-> ![$ [1, 2; 3, 1] $](/files/tex/98b2f3e4134422c8a286f0326fc2f57ca9be2ab7.png)
-> ![$ [1, 2, 3; 2, 1, 1] $](/files/tex/19a9d24510f4551fa45b950aed32efed0636b355.png)
-> ![$ [1, 2, 3; 3, 2, 1] $](/files/tex/b00d573110ba8c2472820409103dd4cbd7bff7cc.png)
-> ![$ [1, 2, 3; 2, 2, 2] $](/files/tex/2aae90b65ae3f116402a1c0143127f80d86acdf0.png)
-> ![$ [1, 2, 3; 1, 4, 1] $](/files/tex/d614399c4078a70fbffb24eb99816a0974423175.png)
-> ![$ [1, 2, 3, 4; 3, 1, 1, 1] $](/files/tex/bf4dd82eaf95bd71a88cb78d3367efa2c7e4c942.png)
-> ![$ [1, 2, 3, 4; 4, 1, 2, 1] $](/files/tex/a4ef5900a9dde9b32311afe7dee2b143f42c405f.png)
-> ![$ [1, 2, 3, 4; 3, 2, 1, 2] $](/files/tex/ff367376ed7bb74fc6207273d7157062b83a1be6.png)
-> ![$ [1, 2, 3, 4; 2, 3, 2, 1] $](/files/tex/81ba2b3608ddba2fc95d32b74b70279f3f5adc5b.png)
, and we now have a fixed point (loop of one array).
such that every set of at least 
collinear points or a subset of ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
and the nonzero derivatives of 
so that every abelian group 
with 
so that the 
has no clique or independent set of size 
. 
-neighbour bootstrap process in the hypercube. 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
-diregular if every vertex has indegree and outdegree at least 
, every 
-diregular oriented graph on at most 
vertices has a Hamilton cycle. 
or 
which divides both 
(see 
(see 
are independent random variables with ![$ \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)
be an abelian group. Is the 
is a finite field with at least 4 elements and 
is an invertible 
matrix with entries in 
which have no coordinates equal to zero such that 
. 
-graph is an 
for every 
with odd size.
for every 
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)
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
? 
and fixed colouring 
of 
with 
colours, there exists 
such that every colouring of the edges of 
contains either 
vertices whose edges are coloured with at most 
colours. 
is a 2-transition system for 
has a compaible decomposition. 
? 
-connected finite graph, is there an assignment of lengths 
to the edges of 
-geodesic cycle is 
be an eulerian tour of 
of co-prime positive integers 
for 
. 
for every filter objects 
and 
and a funcoid 
? 
contains every oriented tree of order 
which are linearly independent over the rational numbers 
, then the extension field 
has transcendence degree of at least 
is a connected cubic graph admitting a 
such that the cubic graph homeomorphic to 
has a 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
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