Weighted Regularity Lemma with Applications

TL;DR

This paper extends the Regularity Lemma to weighted graphs, enabling applications to random graphs and weighted extremal graph theory, while managing exceptional sets effectively.

Contribution

It introduces a weighted version of the Regularity Lemma with techniques to control exceptional sets, broadening its applicability.

Findings

Extended Regularity Lemma applicable to weighted graphs

Applications to random graphs and weighted Erdős-Stone theorem

Methods to handle exceptional sets in weighted graph analysis

Abstract

We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erd\H{o}s-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set.