10 research outputs found
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several
partitioning problems for line and hyperplane arrangements. We prove a
ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l
such that in both line sets, for both halfplanes delimited by l, there are
n^{1/2} lines which pairwise intersect in that halfplane, and this bound is
tight; a centerpoint theorem: for any set of n lines there is a point such that
for any halfplane containing that point there are (n/3)^{1/2} of the lines
which pairwise intersect in that halfplane. We generalize those results in
higher dimension and obtain a center transversal theorem, a same-type lemma,
and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This
is done by formulating a generalization of the center transversal theorem which
applies to set functions that are much more general than measures. Back to
Graph Drawing (and in the plane), we completely solve the open problem that
motivated our search: there is no set of n labelled lines that are universal
for all n-vertex labelled planar graphs. As a side note, we prove that every
set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar
graphs
Semi-algebraic Ramsey numbers
Given a finite point set , a -ary semi-algebraic
relation on is the set of -tuples of points in , which is
determined by a finite number of polynomial equations and inequalities in
real variables. The description complexity of such a relation is at most if
the number of polynomials and their degrees are all bounded by . The Ramsey
number is the minimum such that any -element point set
in equipped with a -ary semi-algebraic relation , such
that has complexity at most , contains members such that every
-tuple induced by them is in , or members such that every -tuple
induced by them is not in .
We give a new upper bound for for and fixed.
In particular, we show that for fixed integers , establishing a subexponential upper bound on .
This improves the previous bound of due to Conlon, Fox, Pach,
Sudakov, and Suk, where is a very large constant depending on and
. As an application, we give new estimates for a recently studied
Ramsey-type problem on hyperplane arrangements in . We also study
multi-color Ramsey numbers for triangles in our semi-algebraic setting,
achieving some partial results
Higher-order Erdos--Szekeres theorems
Let P=(p_1,p_2,...,p_N) be a sequence of points in the plane, where
p_i=(x_i,y_i) and x_1<x_2<...<x_N. A famous 1935 Erdos--Szekeres theorem
asserts that every such P contains a monotone subsequence S of
points. Another, equally famous theorem from the same paper implies that every
such P contains a convex or concave subsequence of points.
Monotonicity is a property determined by pairs of points, and convexity
concerns triples of points. We propose a generalization making both of these
theorems members of an infinite family of Ramsey-type results. First we define
a (k+1)-tuple to be positive if it lies on the graph of a
function whose kth derivative is everywhere nonnegative, and similarly for a
negative (k+1)-tuple. Then we say that is kth-order monotone if
its (k+1)-tuples are all positive or all negative.
We investigate quantitative bound for the corresponding Ramsey-type result
(i.e., how large kth-order monotone subsequence can be guaranteed in every
N-point P). We obtain an lower bound ((k-1)-times
iterated logarithm). This is based on a quantitative Ramsey-type theorem for
what we call transitive colorings of the complete (k+1)-uniform hypergraph; it
also provides a unified view of the two classical Erdos--Szekeres results
mentioned above.
For k=3, we construct a geometric example providing an upper
bound, tight up to a multiplicative constant. As a consequence, we obtain
similar upper bounds for a Ramsey-type theorem for order-type homogeneous
subsets in R^3, as well as for a Ramsey-type theorem for hyperplanes in R^4
recently used by Dujmovic and Langerman.Comment: Contains a counter example of Gunter Rote which gives a reply for the
problem number 5 in the previous versions of this pape
Aligned Drawings of Planar Graphs
Let be a graph that is topologically embedded in the plane and let
be an arrangement of pseudolines intersecting the drawing of .
An aligned drawing of and is a planar polyline drawing
of with an arrangement of lines so that and are
homeomorphic to and . We show that if is
stretchable and every edge either entirely lies on a pseudoline or it has
at most one intersection with , then and have a
straight-line aligned drawing. In order to prove this result, we strengthen a
result of Da Lozzo et al., and prove that a planar graph and a single
pseudoline have an aligned drawing with a prescribed convex
drawing of the outer face. We also study the less restrictive version of the
alignment problem with respect to one line, where only a set of vertices is
given and we need to determine whether they can be collinear. We show that the
problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th
International Symposium on Graph Drawing and Network Visualization (GD 2017
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in ℝ2, there is a line ℓ such that in both line sets, for both halfplanes delimited by ℓ, there are √n lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are √n/3 of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdo{double acute}s-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to graph drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labeled lines that are universal for all n-vertex labeled planar graphs. In contrast, the main result by Pach and Toth (J. Graph Theory 46(1):39-47, 2004), has, as an easy consequence, that every set of n (unlabeled) lines is universal for all n-vertex (unlabeled) planar graphs. © 2012 Springer Science+Business Media, LLC.SCOPUS: ar.jinfo:eu-repo/semantics/publishe