Abstract

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.

1 Introduction

In its broadest sense, the term Ramsey theory refers to any mathematical statement which says that a structure of a given kind is guaranteed to contain a large well-organised substructure. There are examples of such statements in many areas, including geometry, number theory, logic and analysis. For example, a key ingredient in the proof of the Bolzano–Weierstrass theorem in real analysis is a lemma showing that any infinite sequence must contain an infinite monotone subsequence.

A classic example from number theory, proved by van der Waerden [212] in 1927, says that if the natural numbers are coloured in any fixed number of colours then one of the colour classes contains arbitrarily long arithmetic progressions. This result has many generalisations. The most famous, due to Szemerédi [206], says that any subset of the natural numbers of positive upper density contains arbitrarily long arithmetic progressions. This result has many generalisations. The most famous, due to Szemerédi [206], says that any subset of the natural numbers of positive upper density contains arbitrarily long arithmetic progressions.

Abstract

The notion of well quasi-order (wqo) from the theory of ordered sets often arises naturally in contexts where one deals with infinite collections of structures which can somehow be compared, and it then represents a useful discriminator between ‘tame’ and ‘wild’ such classes. In this article we survey such situations within combinatorics, and attempt to identify promising directions for further research. We argue that these are intimately linked with a more systematic and detailed study of homomorphisms in combinatorics.

1 Introduction

In combinatorics, indeed in many areas of mathematics, one is often concerned with classes of structures that are somehow being compared, e.g. in terms of inclusion or homomorphic images. In such situations one is naturally led to consider downward closed collections of such structures under the chosen orderings. The notion of partial well order (pwo), or its mild generalisation well quasi-order (wqo), can then serve to distinguish between the ‘tame’ and ‘wild’ such classes. In this article we will survey the guises in which wqo has made an appearance in different branches of combinatorics, and try to indicate routes for further development which in our opinion will be potentially important and fruitful.

The aim of this article is to identify major general directions in which wqo has been deployed within combinatorics, rather than to provide an exhaustive survey of all the specific results and publications within the topics touched upon. In this section we introduce the notion of wqo, and present what is arguably the most important foundational result, Higman's Theorem. In Section 2 we attempt a broad-brush picture of wqo in combinatorics, linking it to the notion of homomorphism and its different specialised types. The central Sections 3–5 present three ‘case studies’ – words, graphs and permutations – where wqo has been investigated, and draw attention to specific instances of patterns and phenomena already outlined in Section 2. Finally, in Section 6, we reinforce the homomorphism view-point, and explore possible future developments from this angle.

Abstract

We discuss some older and a few recent results related to randomly generated groups. Although most of them are of topological and geometric flavour the main aim of this work is to present them in combinatorial settings.

1 Introduction

For the last half of the century the theory of randomly generated discrete structures has established itself as a vital part of combinatorics. Random graphs and hypergraphs and, more generally, combinatorial, algebraic, and geometric structures generated randomly have been used widely not only to provide numerous examples of objects of exotic properties but also as the way of studying and understanding large non-random systems which often can be decomposed into a small number of pseudorandom parts (see, for instance, Tao [37]). However, until recently, in the theory of random structures as known to combinatorialists random groups have not appeared very frequently (one is tempted to say, sporadically) although Gromov's model of the random group has already been introduced in the early eighties. The main reason was, undoubtedly, the fact that the world of combinatorialists seemed to be quite distant from the land of geometers and topologists and, despite many efforts of a few distinguished mathematicians familiar with both territories, combinatorialists did not believe that one can get basic understanding of the subject without much effort. This landscape has dramatically changed over the last few years. Topological combinatorics (or combinatorial topology) has been developing rapidly; many new projects have been started and a substantial number of articles have been published; combinatorialists have started to use topological terminology and more and more topological works are using advanced combinatorial tools. The aim of this article is just to spread the news. So it is not exactly a survey or even an introduction to this quickly evolving area – the reader who looks for this type of work is referred to a somewhat old but still excellent survey of Ollivier ([34], see also [35]) and the recent paper of Kahle [22].

Abstract

Graph minor theory of Robertson and Seymour is a far reaching generalization of the classical Kuratowski–Wagner theorem, which characterizes planar graphs in terms of forbidden minors. We survey new structural tools and results in the theory, concentrating on the structure of large t-connected graphs, which do not contain the complete graph Kt as a minor.

1 Introduction

Graphs in this paper are finite and simple, unless specified otherwise. A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Numerous theorems in structural graph theory describe classes of graphs which do not contain a fixed graph or a collection of graphs as a minor. A classical example of such a description is the Kuratowski–Wagner theorem [92,93].

Theorem 1.1A graph is planar if and only if it does not contain K5 or K3,3 as a minor.

(We will say that G contains H as a minor, if H is isomorphic to a minor of G, and we will use the notation H ≤ G to denote this. The notation is justified as the minor containment is, indeed, a partial order. We say that G is H-minor free if G does not contain H as a minor.)

Clearly a graph is a forest if and only if it does not contain K3 as a minor. In [16] Dirac proved that a graph does not contain K4 as a minor if and only if it is series-parallel. In [93] Wagner characterizes graphs which do not contain K5 as a minor, as follows.

Theorem 1.2A graph does not contain K5 as a minor if and only if it can be obtained by 0-, 1 and 2 and 3-clique sum operations from planar graphs and V8. (The graph V8 is shown on Figure 1.)