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 ?
Conjecture If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.
Conjecture A total coloring of a graph is an assignment of colors to the vertices and the edges of such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph , , equals the minimum number of colors needed in a total coloring of . It is an old conjecture of Behzad that for every graph , the total chromatic number equals the maximum degree of a vertex in , plus one or two. In other words,
Problem () Find a sufficient condition for a straight -stage graph to be rearrangeable. In particular, what about a straight uniform graph?
Conjecture () Let be a simple regular ordered -stage graph. Suppose that the graph is externally connected, for some . Then the graph is rearrangeable.
Note that the above is a generalization of monotone Galois connections (with and replaced with suprema and infima).
Then we have the following diagram:
What is at the node "other" in the diagram is unknown.
Conjecture "Other" is .
Question What repeated applying of and to "other" leads to? Particularly, does repeated applying and/or to the node "other" lead to finite or infinite sets?
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 .
For a graph , let denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let denote the cardinality of a minimum feedback vertex set (set of vertices so that is acyclic).
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 There exists a fixed constant so that every abelian group has a subset with so that the Cayley graph has no clique or independent set of size .
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
Conjecture Denote by the number of non-Hamiltonian 3-regular graphs of size , and similarly denote by the number of non-Hamiltonian 3-regular 1-connected graphs of size .
Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .
Conjecture For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.
Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as ( times), where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.
Conjecture Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
We say that a set is -universal if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.
Question Does there exist an -universal set of size ?
Conjecture If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.
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is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
. Does the fragment 
have the finite model property? 
and 
be two 
-uniform hypergraph on the same vertex set 
. Does there always exist a partition of 
such that for both 
, at least 
hyperedges of 
meet each of the classes 
and let 
be bases. Then there exist 
each of which is a base. 
denote ''
divides 
''. The mimic function in number theory is defined as follows [1].
divisible by 
, the mimic function, 
, is given by,
. 
and 
, whether there exists a homomorphism from 
for some constant 
? 
of graphs is 
-bounded if there exists a function 
so that every 
satisfies 
. 
, the family of graphs with no induced subgraph isomorphic to ![\[ F_n = 2^{2^{n } } + 1 \]](/files/tex/70ca73d7e82af2fee084a8417e172c58cf78b376.png)
Square-Free?
is a sequence of elements from a multiplicative group of order 
so that 
. 
is an assignment of colors to the vertices and the edges of 
, equals the minimum number of colors needed in a total coloring of 
plus one or two. In other words, ![\[\chi''(G)=\Delta(G)+1\ \ or \ \ \Delta(G)+2.\]](/files/tex/d4ea30e930ec20e02c5a03c6322e6d99a6bdb63a.png)
exist a 
) Find a sufficient condition for a straight 
-stage graph to be rearrangeable. In particular, what about a straight uniform graph? 
) Let 
be a simple regular ordered 
-stage graph. Suppose that the graph 
is externally connected, for some 
. Then the graph 
is rearrangeable. 
paths. 
:
;
. 
and 
replaced with suprema and infima).
. 
and 
to "other" leads to? Particularly, does repeated applying 
, let 
be the statement that given any exact 
, 
, or 
. 
contains a directed path of length 
. 
let 
. 
be a matroid on 
, let 
be a map, put 
and 
. Then 
. The representation function 
for 
is given by the rule 
. We call 
is never 
.
which is slice, is it a ribbon knot?
denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let 
denote the cardinality of a minimum feedback vertex set (set of vertices 
so that 
is acyclic). 
. 
rectangular 
inside an 
? 
. 
into 
so that the graph induced by 
is acyclic for 
, then 
cycles of length 
. 
in the 
-dimensional hypercube? 
so that the 
has no clique or independent set of size 
. 
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)
are matroids on 
for every partition 
of 
with 
which is independent in every 
. 
-manifold has the homotopy type of the 
, is it diffeomorphic to 
the number of non-Hamiltonian 3-regular graphs of size 
, and similarly denote by 
the number of non-Hamiltonian 3-regular 1-connected graphs of size 
? 
be a set of points in the plane. Two points 
and 
in 
there is an integer 
, let 
be the smallest integer 
such that the symmetric group 
on the set of all words of length 
(
is the shuffle permutation defined by 
, and 
is the exchange group consisting of all permutations in 
letters in the words. 
, for all 
if 
if 
. Assume we start with some number 
of the current number. Prove that no matter what the initial number is we eventually reach 
. 
is 
, and all edges are (non-intersecting) straight line segments. 
? 
edges whose deletion destroys every odd cycle. 
can be represented as the sum of an odd prime number and an odd 
, there exist positive integers 
, 
, 
such that 
.
is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease. 
is admissable, and 
is even for every 
, then 
has a faithful cover. 
-connected cubic graph on 
vertices admit a partition into 
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