Applications of Graph and Hypergraph Theory in Geometry

Summary

The aim of this survey is to collect and explain some geometric results whose proof uses graph or hypergraph theory. No attempt has been made to give a complete list of such results. We rather focus on typical and recent examples showing the power and limitations of the method. The topics covered include forbidden configurations, geometric constructions, saturated hypergraphs in geometry, independent sets in graphs, the regularity lemma, and VC-dimension.

1. Introduction

Among n distinct points in the plane the unit distance occurs at most O(n^3/2) times. The proof of this fact uses two things. The first is a theorem from graph theory saying that a graph on n vertices containing no K_2,3 can have at most O(n^3/2) edges. The second is a simple fact from plane geometry: the unit distance graph contains no if K_2,3

This is the first application of graph theory in geometry, and is contained in a short and extremely influential paper of Paul Erdős [1946]. The first application of hypergraph theory in geometry is even earlier: it is the use of Ramsey's theorem in the famous Erdős and Szekeres result from 1935 (see below in the next section). Actually, Erdős and Szekeres proved Ramsey's theorem (without knowing it had been proved earlier) since they needed it for the geometric result.

The aim of this survey is to collect and explain some geometric results whose proof uses graph or hypergraph theory. Such applications vary in depth and difficulty. Often a very simple geometric statement adds an extra condition to the combinatorial structure at hand, which helps in the proof. At other times, the geometry is not so simple but is dictated by the combinatorics of the objects in question.