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.
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 .
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.
Question What is the least integer such that every set of at least points in the plane contains collinear points or a subset of points in general position (no three collinear)?
Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
Question What is the Waring rank of the determinant of a generic matrix?
For simplicity say we work over the complex numbers. The generic matrix is the matrix with entries for . Its determinant is a homogeneous form of degree , in variables. If is a homogeneous form of degree , a power sum expression for is an expression of the form , the (homogeneous) linear forms. The Waring rank of is the least number of terms in any power sum expression for . For example, the expression means that has Waring rank (it can't be less than , as ).
The generic determinant (or ) has Waring rank . The Waring rank of the generic determinant is at least and no more than , see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.
Problem Given two codes , their Tensor Product is the code that consists of the matrices whose rows are codewords of and whose columns are codewords of . The product is said to be robust if whenever a matrix is far from , the rows (columns) of are far from (, respectively).
The problem is to give a characterization of the pairs whose tensor product is robust.
Conjecture Suppose with is a connected cubic graph admitting a -edge coloring. Then there is an edge such that the cubic graph homeomorphic to has a -edge coloring.
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.
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.
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
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?
login/create account
be an integer and let 
be a minor minimal 
. Then either 
minor for some 
or 
and 
positive integers, the (mixed) van der Waerden number 
is the least positive integer 
such that every (red-blue)-coloring of ![$ [1,n] $](/files/tex/661d0acc09fbc5b62f49645a71cf7d831a26563f.png)
admits either a 
, 
. 
for every filter objects 
and 
and a funcoid 
? 
-
is the maximum number of pairwise disjoint edges in 
is the size of the smallest set of vertices which meets every edge, then 
. 
has a dominating set of size 
. 
such that every set of at least 
. Suppose X is an m-by-n integer matrix 
. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
generic matrix? 
for 
. Its determinant is a homogeneous form of degree 
, in 
variables. If 
is a homogeneous form of degree 
, the 
(homogeneous) linear forms. The Waring rank of 
means that 
has Waring rank 
(it can't be less than 
).
generic determinant 
(or 
) has Waring rank 
. The Waring rank of the 
generic determinant is at least 
and no more than 
, see for instance 
. 
is prime and 
, we write 
if 
satisfies 
. The problem of finding such an integer 
(with 
) is the Discrete Log Problem.
, determine whether or not 
, their Tensor Product 
is the code that consists of the matrices whose rows are codewords of 
and whose columns are codewords of 
. The product 
is far from 
is a connected cubic graph admitting a 
-edge coloring. Then there is an edge 
such that the cubic graph homeomorphic to 
has a 
points in the plane in general position contains a subset of 
, it is impossible to cover a square of side greater than 
unit squares. 
. In other words, simplify this formula. 
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. 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
is a tournament of order 
arc-disjoint transitive subtournaments of order 3. 
be a set of ![\[m(A) = - \min_x \sum_{a \in A} \cos(ax).\]](/files/tex/7a772ebd0b7fd98eeab7fda3a5e9674b1dc4c984.png)
What is 
? 
grid. The first player (if any) to occupy a set of 
Drupal
CSI of Charles University