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 Let be an -uniform-partite hypergraph. If is the maximum number of pairwise disjoint edges in , and is the size of the smallest set of vertices which meets every edge, then .
Conjecture If in a bridgeless cubic graph the cycles of any -factor are odd, then , where denotes the oddness of the graph , that is, the minimum number of odd cycles in a -factor of .
Conjecture Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .
The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.
The direct proof may be constructed by correcting all errors an omissions in this draft article.
Direct proof could be better because with it we would get a little more general statement like this:
Conjecture Let be a -separable compact reflexive symmetric funcoid and be a reloid such that \item ; \item .
Problem Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?
If is a finite set of points which is 2-colored, an empty triangle is a set with so that the convex hull of is disjoint from . We say that is monochromatic if all points in are the same color.
Conjecture There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.
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 \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.
Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .
The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.
For and positive integers, the (mixed) van der Waerden number is the least positive integer such that every (red-blue)-coloring of admits either a -term red arithmetic progression or an -term blue arithmetic progression.
Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?
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for reloid 
. 
for every funcoid 
(see below mentioned online article).
denote the pebbling number of a graph 
.
. 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
such that for any 
there is a sequence of tautologies of depth 
Frege system 
but requires exponential size 
proofs. 
for every filter objects 
and 
and a funcoid 
? 
, whether there exists a homomorphism from 
for some constant 
? 
denote the 
is called edge-antipodal if 
whenever 
are antipodal edges. 
and 
which are joined by a monochromatic path. 
-
is the maximum number of pairwise disjoint edges in 
is the size of the smallest set of vertices which meets every edge, then 
. 
-factor are odd, then 
, where 
denotes the oddness of the graph 
in the 
be an abelian group. Is the 
-separable (the same as 
for symmetric transitive) compact funcoid and 
is a uniform space (reflexive, symmetric, and transitive endoreloid) such that 
. Then 
. 
. 
. 
, that every sufficiently large 
-
vertices whose deletion results in a 
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
is disjoint from 
. We say that 
monochromatic empty triangles. 
or 
which divides both 
(see 
(see 
of a locally finite graph 
. 
appears twice, 
appears three times, and 
appears 
times. There are infinite families of numbers known to appear 
times is 
. It is not known whether any number appears more than 
. See 
on an even number of vertices can be decomposed into 
directed paths. 
, the 
, is defined as the sum of all 
by using partitions. 
positive integers, the (mixed) van der Waerden number 
is the least positive integer ![$ [1,n] $](/files/tex/661d0acc09fbc5b62f49645a71cf7d831a26563f.png)
admits either a 
, 
. 
grid. The first player (if any) to occupy a set of 
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