Book contents
- Frontmatter
- Contents
- Preface
- Paul Erdős: The Man and the Mathematician (1913–1996)
- 1 A Selection of Problems and Results in Combinatorics
- 2 Combinatorial Nullstellensatz
- 3 Connectedness, Classes and Cycle Index
- 4 A Tutte Polynomial for Coloured Graphs
- 5 Notes on Sum-Free and Related Sets
- 6 Geometrical Bijections in Discrete Lattices
- 7 On Random Intersection Graphs: The Subgraph Problem
- 8 The Blow-up Lemma
- 9 The Homomorphism Structure of Classes of Graphs
- 10 Problem Collection of the DIMANET Mátraháza Workshop, 22–28 October 1995
8 - The Blow-up Lemma
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- Preface
- Paul Erdős: The Man and the Mathematician (1913–1996)
- 1 A Selection of Problems and Results in Combinatorics
- 2 Combinatorial Nullstellensatz
- 3 Connectedness, Classes and Cycle Index
- 4 A Tutte Polynomial for Coloured Graphs
- 5 Notes on Sum-Free and Related Sets
- 6 Geometrical Bijections in Discrete Lattices
- 7 On Random Intersection Graphs: The Subgraph Problem
- 8 The Blow-up Lemma
- 9 The Homomorphism Structure of Classes of Graphs
- 10 Problem Collection of the DIMANET Mátraháza Workshop, 22–28 October 1995
Summary
Extremal graph theory has a great number of conjectures concerning the embedding of large sparse graphs into dense graphs. Szemerédi's Regularity Lemma is a valuable tool in finding embeddings of small graphs. The Blow-up Lemma, proved recently by Komlós, Sárközy and Szemerédi, can be applied to obtain approximate versions of many of the embedding conjectures. In this paper we review recent developments in the area.
This paper is based on my lectures at the DIMANET Mátraháza Workshop, October 22–28, 1995. On my transparencies, I wrote, ‘For more details see the survey of Komlós–Simonovits in Paul Erdős is 80. Solutions to the conjectures mentioned today will be presented in the Bolyai volume Paul Erdős is 90.’ As you can tell, at that time I expected EP (who was sitting in the front row) to live to be 90 and more. The loss is obvious to all of us, and it will certainly deepen further in time.
Introduction
Our concern in this paper is how Szemerédi's Regularity Lemma can be applied to packing (or embedding) problems. In particular, we discuss a lemma that is a powerful weapon in proving the existence of embeddings of large sparse graphs into dense graphs.
After a brief passage in which we fix the notation, we start in Section 2 by recalling some of the fundamental results and conjectures. Section 3 is about the Regularity Lemma itself; we also demonstrate its power by reconstructing the elegant proof of Ruzsa and Szemeredi for Roth's theorem on arithmetic progressions of length 3.
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- Information
- Recent Trends in CombinatoricsThe Legacy of Paul Erdős, pp. 161 - 176Publisher: Cambridge University PressPrint publication year: 2001