1Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic 2Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, Greece 3Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 625-629
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Martin Doležal; Jan Hladký; Jan Kolář; Themis Mitsis; Christos Pelekis; Václav Vlasák; Martin Doležal; Jan Hladký; Jan Kolář; Themis Mitsis; Christos Pelekis; Václav Vlasák. A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 625-629. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a41/
@article{AMUC_2019_88_3_a41,
author = {Martin Dole\v{z}al and Jan Hladk\'y and Jan Kol\'a\v{r} and Themis Mitsis and Christos Pelekis and V\'aclav Vlas\'ak and Martin Dole\v{z}al and Jan Hladk\'y and Jan Kol\'a\v{r} and Themis Mitsis and Christos Pelekis and V\'aclav Vlas\'ak},
title = { A {Tur\'an-type} theorem for large-distance graphs in {Euclidean} spaces, and related isodiametric problems},
journal = {Acta mathematica Universitatis Comenianae},
pages = {625--629},
year = {2019},
volume = {88},
number = {3},
url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a41/}
}
TY - JOUR
AU - Martin Doležal
AU - Jan Hladký
AU - Jan Kolář
AU - Themis Mitsis
AU - Christos Pelekis
AU - Václav Vlasák
AU - Martin Doležal
AU - Jan Hladký
AU - Jan Kolář
AU - Themis Mitsis
AU - Christos Pelekis
AU - Václav Vlasák
TI - A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems
JO - Acta mathematica Universitatis Comenianae
PY - 2019
SP - 625
EP - 629
VL - 88
IS - 3
UR - http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a41/
ID - AMUC_2019_88_3_a41
ER -
%0 Journal Article
%A Martin Doležal
%A Jan Hladký
%A Jan Kolář
%A Themis Mitsis
%A Christos Pelekis
%A Václav Vlasák
%A Martin Doležal
%A Jan Hladký
%A Jan Kolář
%A Themis Mitsis
%A Christos Pelekis
%A Václav Vlasák
%T A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems
%J Acta mathematica Universitatis Comenianae
%D 2019
%P 625-629
%V 88
%N 3
%U http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a41/
%F AMUC_2019_88_3_a41
A \emph{large-distance graph} is a measurable graph whose vertex set is a measurable subset of $\R^d$, and two vertices are connected by an edge if and only if their distance is larger that 2. We address questions from extremal graph theory in the setting of large-distance graphs, focusing in particular on upper-bounds on the measures of vertices and edges of $K_r$-free large-distance graphs. Our main result states that if $A\subset \R^2$ is a measurable set such that the large-distance graph on $A$ does not contain any complete subgraph on three verticesthen the $2$-dimensional Lebesgue measure of $A$ is at most $2\pi$.
