AVOIDING MULTIPLE REPETITIONS IN EUCLIDEAN SPACES
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
AVOIDING MULTIPLE REPETITIONS IN EUCLIDEAN SPACES
Original language description
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.
Czech name
—
Czech description
—
Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/EF16_027%2F0008360" target="_blank" >EF16_027/0008360: Postdoc@MUNI</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
—
Volume of the periodical
34
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
13
Pages from-to
40-52
UT code for WoS article
000546886700002
EID of the result in the Scopus database
2-s2.0-85079738314