Introduction

The class of graphs containing no induced path of length 3 has many remarkable properties, stemming from the following well-known observation. Recall that an induced subgraph of a graph consists of a subset S of the vertex set together with all edges contained in S.

Proposition. Let G be a finite graph with more than one vertex, containing no induced path of length 3. Then G is connected if and only if its complement is disconnected.

Proof. It is trivial that the complement of a disconnected graph is connected. Moreover, since P3 is self-complementary, the property of containing no induced P3 is self-complementary. So let G be a minimal counterexample: thus, G and Ḡ are connected but, for any vertex v, either G – v or G – v is disconnected. Choose a vertex v and assume, without loss, that G - v is disconnected. Then v is joined to a vertex in each component of G – v.

Preface

Paul Erdős was one of the greatest mathematical figures of the twentieth century. An enormous number of obituaries have appeared since his death, which is very unusual in the world of science.

Paul Erdős had already become a legend in his own lifetime. With a strong character, a clear moral compass and an incredible love of mathematics he created a new mould for a life style. (Will anyone else ever fit it?) This did not require him to pay attention to some of the details of living that most of us deal with. Thus it is not surprising that he developed the eccentricities which are often related by people who knew him and in articles about him – sometimes accurate and sometimes exaggerated. Some of these stories he did not care for, but others he liked to remember, and would retell himself, contributing to the canonization of these anecdotes. As with other passionate geniuses, stories about his eccentricities are a way for writers to show how unusual he was. However, to those who knew him closely, these stories, although amusing, do not in themselves capture the essence of this person, who was so very connected to the world.

Here we shall try to depict Erdős as we saw him.

Mátraháza, 1995

This volume is the Proceedings of the Mátraháza Workshop, one of those workshops which he liked so much, where he felt at home, where he was surrounded by old friends and young mathematicians, all eager to speak with him, to ask him questions, to tell him their results.

The combinatorial workshop ‘Some Trends in Discrete Mathematics’ was held in Matrahaza, Hungary, from 22 to 28 October 1995. The aim of the workshop was to expose connections between distant parts of combinatorial mathematics, such as pure combinatorics, graph theory, combinatorial number theory and random graphs, by bringing together researchers from diverse fields. To emphasize the workshop character of this meeting, we invited many distinguished mathematicians but asked only ten of them to give lectures. (Unfortunately, illness prevented Claude Berge from attending the meeting.) There were no contributed talks, but the lectures were followed by long discussions involving all the participants: these sessions played a crucial role in the success of the workshop. A tangible result of these evening discussions is the Cameron–Erdős paper in this volume.

A highlight of the volume is the paper Paul Erdős was writing on the eve of his sudden death in Warsaw on 20 September, 1996. This paper had no title and, except for light editing, this very special manuscript is published as he left it. The other eight papers of this issue are surveys and research papers written by the invited speakers and their collaborators.

We want to thank all the participants of the workshop for their contribution to its success. We are also grateful to DIMANET and its main coordinator, Professor Walter Deuber, for providing the financial support that made the workshop possible. We wish to express our sincere thanks to Béla Bollobás who made it possible to publish these papers in a special issue of Combinatorics, Probability and Computing.

Introduction

Basic definitions

Throughout this paper, we shall consider the set of finite multigraphs, with loops allowed. Usually, we shall call an element of a graph, but sometimes we shall write multigraph for emphasis.

The Tutte polynomial, or dichrornate of [34], is an isomorphism-invariant function which arises in many different ways. We shall consider four different definitions of the Tutte polynomial.

The first definition, due to Tutte [34], is based on a spanning tree expansion. We take an order ø on E(G), and, for each spanning tree S of G, use ø and S to classify the edges of G into four types. We then assign a weight to each edge, depending on its type, and multiply these weights to find the weight of S.

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.