A Turán-Type Theorem for Large-Distance Graphs in Euclidean Spaces, and Related Isodiametric Problems

Abstract

We consider the problems ofmaximizing the 2d-dimensional Lebesgue measure of the edge set as well as the d-dimensional Lebesgue measure of the vertex set of a large-distance graph in the d-dimensional Euclidean space that contains no copies of a complete graph on k vertices.

Keywords

• Turán’s theorem
• Isodiametric problem
• Distance graph