%A Doležal, Martin
%A Hladký, Jan
%A Kolář, Jan
%A Mitsis, Themis
%A Pelekis, Christos
%A Vlasák, Václav
%D 2019
%T A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems
%K
%X 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 $2\pi$.
%U http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1300
%J Acta Mathematica Universitatis Comenianae
%0 Journal Article
%P 625-629%V 88
%N 3
%@ 0862-9544
%8 2019-08-01