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 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 .
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 .
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Conjecture Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.
Proposition It is easy to prove that is the infinitely small right neighborhood filter of point .
If proved true, the conjecture then can be generalized to a wider class of posets.
Conjecture There exists a fixed constant (probably suffices) so that every graft with minimum -cut size at least contains a -join packing of size at least .
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does there exists a 
such that every graph with average degree smaller than 
has chromatic number at most 
.
such that for every graph 
, ![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
vertices has an edge colouring such that:
cycles. 
, there exists a constant 
, so that every graph 
. 
-regular graph, where 
, then 
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 
. 
exist a 
-chromatic graph on at most 
vertices, then 
. 
. 
denote the set of all permutations of ![$ [n]=\set{1,2,\ldots,n} $](/files/tex/470bd09059264370b76b4da90b77dc370c7e0e7c.png)
. Let 
and 
denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in 
. Let 
denote the set of subsequences of 
denote 
.
is called a Roller Coaster permutation if 
. Let 
be the set of all Roller Coaster permutations in 
,
, then 
. \item If 
, then 
with 
. 
,
is odd for 
. \item If 
for all 
be an integer and let 
denote the least integer 
, let 
be the smallest integer 
such that the symmetric group 
on the set of all words of length 
(
is the shuffle permutation defined by 
, and 
is the exchange group consisting of all permutations in 
letters in the words. 
, for all 
be the complete funcoid corresponding to the usual topology on extended real line ![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)
. Let 
be the order on this set. Then 
is a complete funcoid. 
is the infinitely small right neighborhood filter of point ![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
. 
(probably 
suffices) so that every graft with minimum 
-cut size at least 
. 
in 
such that the fundamental group of the complement has an unsolvable word problem? 
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