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

Abstract

Given a measurable set $A\subset \mathbb R^d$ we consider the large-distance graph $\mathcal {G}_A$, on the ground set A, in which each pair of points from A whose distance is bigger than 2 forms an edge. We consider the problems of maximizing 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. The former problem may be seen as a continuous analogue of Turán’s classical graph theorem, and the latter as a “graph-theoretic” analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel’s theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest.