# A Tur\'{a}n-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems

## TL;DR

This paper extends Turán-type theorems to large-distance graphs in Euclidean spaces, establishing bounds on edge measures and vertex measures related to forbidden complete subgraphs, and connects to classical isodiametric problems.

## Contribution

It introduces a continuous analogue of Turán's theorem for large-distance graphs and proves an isodiametric inequality in an annulus, advancing geometric extremal graph theory.

## Key findings

- Established a Turán-type bound for large-distance graphs in Euclidean spaces.
- Proved an isodiametric inequality in an annulus relevant to the problem.
- Connected extremal graph theory with classical geometric inequalities.

## 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 Tur\'an'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.