Conjecture There is an integer-valued function such that if is any -connected graph and and are any two vertices of , then there exists an induced path with ends and such that is -connected.
Problem Find a constant such that for any there is a sequence of tautologies of depth that have polynomial (or quasi-polynomial) size proofs in depth Frege system but requires exponential size proofs.
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 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 A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?
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?
Conjecture For all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order .
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 .
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in 
such that the fundamental group of the complement has an unsolvable word problem? 
has chromatic number at most 
.
![\[ M_n = 2^p - 1 \]](/files/tex/eb18a56e5c2e8b1be6ac733d217c0c1f1a47a94e.png)
that is prime.
with no 
with 
so that 
has a 
. 
such that if 
and 
are any two vertices of 
with ends 
is 
-connected. 
has a compaible decomposition. 
be an elliptic curve over a number field 
. Then the order of the zeros of its 
-function, 
, at 
is the Mordell-Weil rank of 
. 
there is a sequence of tautologies of depth 
Frege system 
but requires exponential size 
proofs. 
there is an integer 
such that every set of at least 
collinear points or an empty hexagon. 
rectangular 
-dimensional boxes each of which has side lengths 
inside an 
? 
of 
of 
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)
. ![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
. 
we have 
. Then 
is graphic. 
be a 
. Is there a self-homeomorphism 
of 
greater than 
so that 
does not intersect 
be a prime natural number. Find all primes 
, such that 
.
-flow 
on 
of 
from the edge set of 
, for all 
, and 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
is arbitrarily large? 
, there exists a constant 
, so that every graph 
. 
be an integer and let 
denote the least integer 
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