10 research outputs found
Erdos-Szekeres-type statements: Ramsey function and decidability in dimension 1
A classical and widely used lemma of Erdos and Szekeres asserts that for
every n there exists N such that every N-term sequence a of real numbers
contains an n-term increasing subsequence or an n-term nondecreasing
subsequence; quantitatively, the smallest N with this property equals
(n-1)^2+1. In the setting of the present paper, we express this lemma by saying
that the set of predicates Phi={x_1<x_2,x_1\ge x_2}$ is Erdos-Szekeres with
Ramsey function ES_Phi(n)=(n-1)^2+1.
In general, we consider an arbitrary finite set Phi={Phi_1,...,Phi_m} of
semialgebraic predicates, meaning that each Phi_j=Phi_j(x_1,...,x_k) is a
Boolean combination of polynomial equations and inequalities in some number k
of real variables. We define Phi to be Erdos-Szekeres if for every n there
exists N such that each N-term sequence a of real numbers has an n-term
subsequence b such that at least one of the Phi_j holds everywhere on b, which
means that Phi_j(b_{i_1},...,b_{i_k}) holds for every choice of indices
i_1,i_2,...,i_k, 1<=i_1<i_2<... <i_k<= n. We write ES_Phi(n) for the smallest N
with the above property.
We prove two main results. First, the Ramsey functions in this setting are at
most doubly exponential (and sometimes they are indeed doubly exponential): for
every Phi that is Erd\H{o}s--Szekeres, there is a constant C such that
ES_Phi(n) < exp(exp(Cn)). Second, there is an algorithm that, given Phi,
decides whether it is Erdos-Szekeres; thus, one-dimensional
Erdos-Szekeres-style theorems can in principle be proved automatically.Comment: minor fixes of the previous version. to appear in Duke Math.
Extending the Centerpoint Theorem to Multiple Points
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.
We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n)
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Tverberg's theorem is 50 Years Old: A survey
This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. © 2018 American Mathematical Society
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Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing abstracts of the presentations and a summary of the problem session
One-sided epsilon-approximants
Suppose and are sets in such that every convex set containing α-fraction of points P contains at least -fraction of points of , for every . In such a case, set is called a one-sided -approximant to . We show that every admits a one-sided -approximant of size depending only on and on . (Joint work with Gariel Nivasch.)Non UBCUnreviewedAuthor affiliation: Carnegie Mellon UniversityFacult