Question Is the binary affine cube the only 3-connected matroid for which equality holds in the bound where is the circumference (i.e. largest circuit size) of ?
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.
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 Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .
Conjecture Is it possible to color edges of the complete graph using colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
Equivalently: is the star chromatic index of linear in ?
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 If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.
An -factor in a graph is a set of vertex-disjoint copies of covering all vertices of .
Problem Let be a fixed positive real number and a fixed graph. Is it NP-hard to determine whether a graph on vertices and minimum degree contains and -factor?
Problem Does the following equality hold for every graph ?
The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number, we minimize the number of pairs of edges that cross.
For any simple digraph , we let be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and be the size of the smallest feedback edge set.
Conjecture If is a simple digraph without directed cycles of length , then .
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.
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 For every prime , there is a constant (possibly ) so that the union (as multisets) of any bases of the vector space contains an additive basis.
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 .
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be a prime natural number. Find all primes 
, such that 
.
the only 3-connected matroid for which equality holds in the bound 
where 
is the circumference (i.e. largest circuit size) of 
? 
. 
appears once in Pascal's triangle, 
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 
is a 
-separable (the same as 
for symmetric transitive) compact funcoid and 
is a uniform space (reflexive, symmetric, and transitive endoreloid) such that 
. Then 
. 
. 
. 
be a graph, 
a countable end of 
an infinite set of pairwise disjoint 
of pairwise disjoint 
using 
colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?
? 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
, then there is a 
-factor in a graph 
be a fixed positive real number and 
contains and ![\[ \text{pair-cr}(G) = \text{cr}(G) \]](/files/tex/8cece1e00bb0e9fc122e0a5cad0dab2681cf33a4.png)
of a graph 
, we minimize the number of pairs of edges that cross. 
-
(mod 
) for every vertex 
. 
be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and 
be the size of the smallest 
, then 
. 
of a locally finite graph 
, every digraph with minimum out-degree at least 
contains 
the following inequality holds 
? 
there is a sequence of tautologies of depth 
Frege system 
but requires exponential size 
proofs. 
(possibly 
) so that the union (as multisets) of any 
contains an additive basis. 
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
for 
. 
be an integer. For every integer 
, there exists an integer 
such that for every digraph 
, either 
, or there exists a set 
of at most 
vertices such that 
has no directed cycles of length at least 
are invertible 
matrices with entries in 
for a prime 
submatrix 
of ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
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