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 integer. For every integer , there exists an integer such that for every digraph , either has a pairwise-disjoint directed cycles of length at least , or there exists a set of at most vertices such that has no directed cycles of length at least .
Let be a set, be the set of filters on ordered reverse to set-theoretic inclusion, be the set of principal filters on , let be an index set. Consider the filtrator .
Conjecture If is a completary multifuncoid of the form , then is a completary multifuncoid of the form .
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?
Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.
Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?
2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?
3. Finally, what could one say about higher dimensional analogs of this question?
Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n
Conjecture For all positive integers and , there exists an integer such that every graph of average degree at least contains a subgraph of average degree at least and girth greater than .
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.
Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .
Question Does the Blatter-Specker Theorem hold for ternary relations.
Begin with the generating function for unrestricted partitions:
(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...
Now change some of the plus signs to minus signs. The resulting series will have coefficients congruent, mod 2, to the coefficients of the generating series for unrestricted partitions. I conjecture that the signs may be chosen such that all the coefficients of the series are either 1, -1, or zero.
Conjecture Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .
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-connected cubic planar graph can be decomposed into two Hamilton cycles. 
such that for any 
there is a sequence of tautologies of depth 
Frege system 
but requires exponential size 
proofs. 
of 
is the minimum number of crossings in all drawings of 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, either 
pairwise-disjoint directed cycles of length at least 
, or there exists a set 
of at most 
vertices such that 
has no directed cycles of length at least 
is a sequence of elements from a multiplicative group of order 
so that 
. 
and 
be a set, 
be the set of filters on 
be the set of principal filters on 
.
, then 
is a completary multifuncoid of the form 
. ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
and the nonzero derivatives of 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
. 
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. 
be a class of finite relational structures. We denote by 
the number of structures in 
. For any class 
, the function 
are simple finite graphs, then 
. 
is the 
.
for some constant 
? 
is a graph, 
is admissable, and 
is even for every 
, then 
has a faithful cover. 
to be the maximum degree, 
to be the size of the largest 
to be the chromatic number of 
for every graph 
-minor. 
be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
are prime.
rectangular 
inside an 
? 
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