A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems
2019
Abstract
The authors discuss large-distance graphs on measurable subsets of Rd, principally considering d=2. That is, given a d-dimensional vertex set A, edges are defined to join pairs of vertices whose distance exceeds 2.
The authors provide an extended abstract presenting their primary result. If A is any 2-dimensional measurable set such that the large-distance graph on A is triangle-free, then the Lebesgue measure of A is at most 2π and that of its edge set is at most 2π2.