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Unsplittable Coverings in the Plane

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Extremal Distance And Conformal Radius Of A Cle4 Loop

Juhan Aru, Titus Lupu

Consider CLE4 in the unit disk, and let l be the loop of the CLE4 surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by l. We complement their result by determining ...

INST MATHEMATICAL STATISTICS-IMS2022

Forbidden induced subposets of given height

Istvan Tomon

Let P be a partially ordered set. The function La* (n, P) denotes the size of the largest family F subset of 2([n]) that does not contain an induced copy of P. It was proved by Methuku and Palvolgyi that there exists a constant C-P (depending only on P) su ...

ACADEMIC PRESS INC ELSEVIER SCIENCE2019

We consider sets L = {l(1),..., l(n)} of n labeled lines in general position in R-3, and study the order types of point sets {p(1),..., p(n)} that stem from the intersections of the lines in L with (directed) planes Pi, not parallel to any line of L, that ...

ELSEVIER SCIENCE BV2019

Counting Intersecting and Pairs of Cross-Intersecting Families

Andrei Kupavskii

A family of subsets of {1, ... , n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdos, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1, ... , n}. In this ...

Cambridge Univ Press2018

Unsplittable coverings in the plane

János Pach

A system of sets forms an m-fold covering of a set X if every point of X belongs to at least m of its members. A 1-fold covering is called a covering. The problem of splitting multiple coverings into several coverings was motivated by classical density est ...

Academic Press Inc Elsevier Science2016

Suppose k is a positive integer and X is a k-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most k sets. Suppose there is a function f(n) = o(n(2)) with the property that any ...

Cambridge Univ Press2016

Curvature-Controlled Defect Localization in Elastic Surface Crystals

Pedro Miguel Nunes Pereira de Almeida Reis

We investigate the influence of curvature and topology on crystalline dimpled patterns on the surface of generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadrati ...

American Physical Society2016

Comment on "Room Temperature Electrically Injected Polariton Laser"

Benoît Marie Joseph Deveaud

In this Comment I challenge the main conclusions of Ref. [1]. The main reason for this Comment is the large number of questions on the details of the experiment that led me to request the raw data. I have obtained two sets of raw data. After a long discuss ...

American Physical Society2016

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line L if the intersection of any member with L is a nonempty segment. It is proved that the intersection graphs of simple families of com ...

Springer2014

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Bartosz Maria Walczak, Michal Lason

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the ...

Springer US2013