AVOIDING MULTIPLE REPETITIONS IN EUCLIDEAN SPACES
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14330%2F20%3A00115528" target="_blank" >RIV/00216224:14330/20:00115528 - isvavai.cz</a>
Výsledek na webu
<a href="https://doi.org/10.1137/18M1180347" target="_blank" >https://doi.org/10.1137/18M1180347</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/18M1180347" target="_blank" >10.1137/18M1180347</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
AVOIDING MULTIPLE REPETITIONS IN EUCLIDEAN SPACES
Popis výsledku v původním jazyce
We study colorings of Euclidean spaces avoiding specified patterns on straight lines. This extends the seminal work of Thue on avoidability properties of sequences to continuous, higher dimensional structures. We prove that every space R^d has a 2-coloring such that no sequence of colors derived from collinear points separated by unit distance consists of more than r(d) identical blocks. In case of the plane we show that r(2) <= 43. We also consider more general patterns and give a sufficient condition for a pattern to be avoided in the plane. This supports a general Pattern Avoidance Conjecture in Euclidean spaces. The proofs are based mainly on the probabilistic method, but additional tools are forced by the geometric nature of the problem. We also consider similar questions for general geometric graphs in the plane. In the conclusion of the paper, we pose several conjectures alluding to some famous open problems in Euclidean Ramsey Theory.
Název v anglickém jazyce
AVOIDING MULTIPLE REPETITIONS IN EUCLIDEAN SPACES
Popis výsledku anglicky
We study colorings of Euclidean spaces avoiding specified patterns on straight lines. This extends the seminal work of Thue on avoidability properties of sequences to continuous, higher dimensional structures. We prove that every space R^d has a 2-coloring such that no sequence of colors derived from collinear points separated by unit distance consists of more than r(d) identical blocks. In case of the plane we show that r(2) <= 43. We also consider more general patterns and give a sufficient condition for a pattern to be avoided in the plane. This supports a general Pattern Avoidance Conjecture in Euclidean spaces. The proofs are based mainly on the probabilistic method, but additional tools are forced by the geometric nature of the problem. We also consider similar questions for general geometric graphs in the plane. In the conclusion of the paper, we pose several conjectures alluding to some famous open problems in Euclidean Ramsey Theory.
Klasifikace
Druh
J<sub>imp</sub> - Článek v periodiku v databázi Web of Science
CEP obor
—
OECD FORD obor
10101 - Pure mathematics
Návaznosti výsledku
Projekt
<a href="/cs/project/EF16_027%2F0008360" target="_blank" >EF16_027/0008360: Postdoc@MUNI</a><br>
Návaznosti
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Ostatní
Rok uplatnění
2020
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
—
Svazek periodika
34
Číslo periodika v rámci svazku
1
Stát vydavatele periodika
US - Spojené státy americké
Počet stran výsledku
13
Strana od-do
40-52
Kód UT WoS článku
000546886700002
EID výsledku v databázi Scopus
2-s2.0-85079738314