Conjecture Can the approximation ratio be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than -hardness?
Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .
Conjecture [2] Let be a graph with list chromatic number and . Then
Conjecture For every , the sequence in consisting of copes of and copies of has the fewest number of distinct subsequence sums over all zero-free sequences from of length .
Problem Is there a minimum integer such that the vertices of any digraph with minimum outdegree can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least ?
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 For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .
Conjecture \item If is a countable connected graph then its third power is hamiltonian. \item If is a 2-connected countable graph then its square is hamiltonian.
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 .
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.
Conjecture Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.
A -page book embedding of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.
One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.
The book thickness of , denoted by bt is the minimum integer for which there is a -page book embedding of .
Let be the graph obtained by subdividing each edge of exactly once.
Conjecture There is a function such that for every graph ,
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 family of multifuncoids such that each is of the form where is an index set for every and is a set for every . Let every for some multifuncoid of the form regarding the filtrator . Let is a graph-composition of (regarding some partition and external set ). Then there exist a multifuncoid of the form such that regarding the filtrator .
Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical product:
\item Product morphism is defined similarly to the category of topological spaces. \item Product object is the sub-atomic product. \item Projections are sub-atomic projections.
See details, exact definitions, and attempted proofs here.
Problem Let and be two -uniform hypergraph on the same vertex set . Does there always exist a partition of into classes such that for both , at least hyperedges of meet each of the classes ?
Question \item Does hold over graphs of bounded tree-width? \item Is included in over graphs? \item Does have a 0-1 law? \item Are properties of Hanf-local? \item Is there a logic (with an effective syntax) that captures ?
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(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than 
-hardness? 
be a simple graph, and for every list assignment 
let 
be the maximum number of vertices of 
, where the minimum is taken over all list assignments 
for all 
. 
and 
. Then ![\[\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.\]](/files/tex/47be18e956355dd433b88b66eabf01a9e3ed5f61.png)
such that every planar graph has obstacle number at most 
vertices has an edge colouring such that:
, the sequence in 
consisting of 
copes of 
and 
copies of 
has the fewest number of distinct subsequence sums over all zero-free sequences from 
. 
such that the vertices of any digraph with minimum outdegree 
can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least 
points in the plane in general position contains a subset of 
, there is a constant 
(possibly 
) so that the union (as multisets) of any 
contains an additive basis. 
, let 
be the statement that given any exact 
-colored countably infinite complete subgraph. Then 
, 
, or 
. 
or 
which divides both 
(see 
(see 
so that the power series 
is bounded for all 
. 
be a class of finite relational structures. We denote by 
the number of structures in 
. For any class 
, the function 
which is slice, is it a ribbon knot?
be the unique integer (with respect to a fixed 
) such that
is a prime iff ![$$ \displaystyle \sum_{i=1}^{\left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor} r_i = \left \lfloor \frac{\sqrt[3]p}{2} \right \rfloor $$](/files/tex/99af565f4cc4d3bab11eb3fbf54f78626678d484.png)
is 2-large, then 
of 
and a (non-proper) 
such that edges with the same colour do not cross with respect to 
for some edges 
, then 
and 
receive distinct colours. 
is the minimum integer 
be the graph obtained by subdividing each edge of 
such that for every graph 
-factor are odd, then 
, where 
denotes the oddness of the graph 
is a family of multifuncoids such that each 
is of the form 
where 
is an index set for every 
and 
is a set for every 
. Let every 
for some multifuncoid 
of the form 
regarding the filtrator 
. Let 
is a graph-composition of 
). Then there exist a multifuncoid 
of the form 
such that 
regarding the filtrator 
. 
be a matroid of rank 
, and for 
let 
be the number of closed sets of rank 
is unimodal. 
denote the pebbling number of a graph 
. 
and 
be two 
. Does there always exist a partition of 
such that for both 
, at least 
hyperedges of 
meet each of the classes 
hold over graphs of bounded tree-width? \item Is 
included in 
over graphs? \item Does 
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