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)?
Problem has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .
If is a finite set of points which is 2-colored, an empty triangle is a set with so that the convex hull of is disjoint from . We say that is monochromatic if all points in are the same color.
Conjecture There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.
Conjecture Let be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists such that is an element of at least half the members of .
Conjecture If is the adjacency matrix of a -regular graph, then there is a symmetric signing of (i.e. replace some entries by ) so that the resulting matrix has all eigenvalues of magnitude at most .
Conjecture For all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order .
The crossing number of is the minimum number of crossings in all drawings of in the plane.
The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.
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be a loopless multigraph. Then the edge chromatic number of 
of 
denote the graph with vertex set consisting of all lines through the origin in 
and two vertices adjacent in 
? 
vertices can be decomposed into at most 
cycles. 
-regular graph, where 
, then 
can be replaced by a smaller value, we can also write the conjecture as, for any positive 
(probably 
suffices) so that every embedded (loopless) graph with edge-width 
has a 5-local-tension. 
is for every sets 
and 
such that every set of at least 
such that every planar graph has obstacle number at most 
has the homotopy-type of a product space 
where 
is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of 
, there is only a finite number of good-edge-labeling critical graphs with average degree less than 
for 
. 
-
of 
so that 
is 
-edge-connected and 
is 
-edge-connected? 
, determine whether or not 
is a finite set of points which is 2-colored, an empty triangle is a set 
with 
so that the convex hull of 
is disjoint from 
. We say that 
monochromatic empty triangles. 
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists 
such that 
satisfies 
. 
denote the 
? 
is the adjacency matrix of a 
-regular graph, then there is a symmetric signing of 
entries by 
) so that the resulting matrix has all eigenvalues of magnitude at most 
. 
contains 
such that every digraph of minimum outdegree at least 
is the graph whose vertices are all binary sequences of length 
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