A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems A Turán-type theorem, and related isodiametric problems

Abstract

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 $2\pi$.