# The de Bruijn-Erd\"os-Hanani theorem

## TL;DR

This paper reviews the de Bruijn-Erd"os-Hanani theorem in finite geometry, discussing its various proofs, their discovery processes, and providing insights into their development.

## Contribution

It offers a comprehensive account of the theorem's proofs, including original and alternative methods, enhancing understanding of their historical and mathematical context.

## Key findings

- Multiple proofs of the theorem are analyzed.
- Insights into the discovery process of the proofs are provided.
- The paper clarifies the theorem's significance in finite geometry.

## Abstract

The paper is devoted to a somewhat idiosyncratic account of the theorem of de Bruijn-Erd\"{o}s and Hanani from the combinatorics of finite geometries and its various proofs. Among the proofs discussed are the original proofs by de Bruijn--Erd\"{o}s and Hanani and few others. It is explained how one of the proofs was discovered, and how another one could have been discovered.