A Pythagoras proof of Szemer\'edi's regularity lemma

TL;DR

This paper presents a concise proof of Szemerédi's regularity lemma using elementary Euclidean geometry, simplifying the traditional proof by leveraging Pythagoras' theorem to reduce technical complexities.

Contribution

The paper introduces a novel geometric approach to prove Szemerédi's regularity lemma, streamlining the proof process with elementary Euclidean geometry techniques.

Findings

Simplified proof of Szemerédi's regularity lemma

Reduction of technical complexities in the proof

Application of Pythagoras' theorem in combinatorial proof

Abstract

We give a short proof of Szemer\'edi's regularity lemma, based on elementary Euclidean geometry. The general line of the proof is that of the standard proof (in fact, of Szemer\'edi's original proof), but most technicalities are swallowed by applying Pythagoras' theorem.