Results 1  10
of
13
Tight Lower Bounds for the Size of EpsilonNets
"... According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O () ..."
Abstract

Cited by 22 (1 self)
 Add to MetaCart
According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VCdimension admits an εnet of size O ()
Erdős–Szekerestype statements: Ramsey function and decidability in dimension 1
, 2012
"... A classical and widely used lemma of Erdős and Szekeres asserts that for every n there exists N such that every Nterm sequence a of real numbers contains an nterm increasing subsequence or an nterm nonincreasing subsequence; quantitatively, the smallest N with this property equals (n − 1)2 + 1. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
A classical and widely used lemma of Erdős and Szekeres asserts that for every n there exists N such that every Nterm sequence a of real numbers contains an nterm increasing subsequence or an nterm nonincreasing 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 Φ = {x1 < x2, x1 ≥ x2} is Erdős–Szekeres with Ramsey function ESΦ(n) = (n − 1)2 + 1. In general, we consider an arbitrary finite set Φ = {Φ1,...,Φm} of semialgebraic predicates, meaning that each Φj = Φj(x1,..., xk) is a Boolean combination of polynomial equations and inequalities in some number k of real variables. We define Φ to be Erdős– Szekeres if for every n there exists N such that each Nterm sequence a of real numbers has an nterm subsequence b such that at least one of the Φj holds everywhere on b, which means that Φj(bi1,..., bik) holds for every choice of indices i1, i2,..., ik, 1 ≤ i1 < i2 < · · · < ik ≤ n. We write ESΦ(n) for the smallest N with the above property. We prove two main results. First, the Ramsey functions in this setting are at most
Small Strong Epsilon Nets
"... In this paper, we initiate the study of small strong ϵnets and prove bounds for axisparallel rectangles, half spaces, strips and wedges. We also give some improved bounds for small weak ϵnets. 1 ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
In this paper, we initiate the study of small strong ϵnets and prove bounds for axisparallel rectangles, half spaces, strips and wedges. We also give some improved bounds for small weak ϵnets. 1
New ɛNet Constructions
"... In this paper, we give simple and intuitive constructions to obtain linear size ɛnets for αfat wedges, translations and rotations of a quadrant and axisparallel threesided rectangles in R 2. We also give new constructions using elementary geometry to obtain linear size weak ɛnet for dhypercube ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper, we give simple and intuitive constructions to obtain linear size ɛnets for αfat wedges, translations and rotations of a quadrant and axisparallel threesided rectangles in R 2. We also give new constructions using elementary geometry to obtain linear size weak ɛnet for dhypercubes and disks in R 2. 1