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

Abstract

Given a measurable set A ⊂ R d we consider the large-distance graph 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.Keywords Turán's theorem · isodiametric problem · distance graph Mathematics Subject Classification (2010) 05C63 · 51K99 · 05C35 · 51M16 1 Prologue, related work and main resultsLet us begin with a folklore result of Turán which pertains to graphs that contain no complete graph (also called a clique) on k vertices. Given a graph G = (V, E), we denote by |V | and |E| the number of its vertices and edges, respectively. Theorem 1.1 (Turán [22]) Let G = (V, E) be a graph which does not contain a complete graph on k vertices. Then |E| ≤ 1 2 1 − 1 k−1 · |V | 2 .