Very important

This article is accompanied by the Maple packages

• http://www.math.rutgers.edu/~zeilberg/tokhniot/RITSUF,

• http://www.math.rutgers.edu/~zeilberg/tokhniot/RITSUFwt,

• http://www.math.rutgers.edu/~zeilberg/tokhniot/ARGF,

to be described below. In fact, more accurately, this article accompanies these packages, written by DZ and the many output files, discovering and proving deep enumeration theorems, done by SBE, that are linked to from the webpage of this article: http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ritsuf.html.

How it all started: April 5, 2012

During one of the Rutgers University Experimental Mathematics Seminar dinners, the name of Don Knuth came up, and two of the participants, David Nacin, who was on sabbatical from William Patterson University, and first-year graduate student Patrick Devlin, mentioned that they recently solved a problem that Knuth proposed in Mathematics Magazine [5]. The problem was:

1868. Proposed by Donald E. Knuth, Stanford University, Stanford, California.

Let n ≥ 2 be an integer. Remove the central (n − 2)2 squares from an (n + 2) × (n + 2) array of squares. In how many ways can the remaining squares be covered with 4n dominoes?

Introduction

Reconfiguration problems are combinatorial problems in which we are given a collection of configurations, together with some transformation rule(s) that allows us to change one configuration to another. A classic example is the so-called 15-puzzle (see Figure 1): 15 tiles are arranged on a 4 × 4 grid, with one empty square; neighbouring tiles can be moved to the empty slot. The normal aim is, given an initial configuration, to move the tiles to the position with all numbers in order (right-hand picture in Figure 1). Readers of a certain age may remember Rubik’s cube and its relatives as examples of reconfiguration puzzles (see Figure 2).

Introduction

Optimization problems such as coloring a graph, or finding the size of a largest clique or stable set are NP-hard in general, but become polynomially solvable when some configurations are excluded. On the other hand they remain difficult even when seemingly quite a lot of structure is imposed on an input graph. For example, determining whether a graph is 3-colorable remains NP-complete for triangle-free graphs with maximum degree 4 [92].

Introduction

Groups are often studied in terms of their action on the elements of a set or on particular objects within a structure. Examples of such situations are abundant and we mention here just a few. Since Cayley's time we know that every group can be viewed as a group of permutations on a set. The study of group actions on vector spaces gave rise to the vast area of representation theory. Investigation of automorphism groups of field extensions generated challenges such as the Inverse Galois Problem. In low-dimensional topology, group actions on trees and on graphs in general led to important findings regarding growth of groups.

Introduction

Galois spaces, that is affine and projective spaces of dimension N > 2 defined over a finite (Galois) field Fq, are well known to be rich in nice geometric, combinatorial and group-theoretic properties that have also found wide and relevant applications in several branches of combinatorics, especially to design theory and graph theory, as well as in more practical areas, notably coding theory and cryptography.

The systematic study of Galois spaces was initiated in the late 1950's by the pioneering work of B. Segre [77]. The trilogy [53, 55, 58] covers the general theory of Galois spaces including the study of objects which are linked to linear codes. Typical such objects are plane arcs and their generalizations, especially caps, saturating sets and arcs in higher dimensions, whose code-theoretic counterparts are distinguished types of error-correcting and covering linear codes, such as MDS codes. Their investigation has received a great stimulus from coding theory, especially in the last decades; see the survey papers [56, 57].

The Twenty-Fourth British Combinatorial Conference was organised by Royal Holloway, University of London. It was held in Egham, Surrey in July 2013. The British Combinatorial Committee had invited nine distinguished combinatorialists to give survey lectures in areas of their expertise, and this volume contains the survey articles on which these lectures were based.

In compiling this volume we are indebted to the authors for preparing their articles so accurately and professionally, and to the referees for their rapid responses and keen eye for detail. We would also like to thank Roger Astley and Sam Harrison at Cambridge University Press for their advice and assistance.

Finally, without the previous efforts of editors of earlier Surveys and the guidance of the British Combinatorial Committee, the preparation of this volume would have been daunting: we would like to express our thanks for their support.

Introduction

The notion of permutation patterns is implicit in the literature a long way back, which is no surprise given that permutations are a natural object in many branches of mathematics, and because patterns of various sorts are ubiquitous in any study of discrete objects. In recent decades the study of permutation patterns has become a discipline in its own right, with hundreds of published papers. This rapid development has not only led to myriad new results, but also, and more interestingly, spawned several different research directions in the last few years. Also, many connections have been discovered between permutation patterns and other research areas, both inside and outside of combinatorics, showcasing the fundamental nature of patterns in permutations and other kinds of words.

Introduction

For the last thirteen years we have been involved in a collaborative project to generalise the results of the Graph Minors Project of Robertson and Seymour to matroids representable over finite fields. The banner theorems of the Graph Minors Project are that graphs are well-quasi-ordered under the minor order [34] (that is, in any infinite set of graphs there is one that is isomorphic to a minor of another) and that for each minor-closed class of graphs there is a polynomial-time algorithm for recognising membership of the class [32]. We are well on track to extend these theorems to the class of F-representable matroids for any finite field F.

It is important to point out here that day-to-day work along this track does not concern well-quasi-ordering or minor testing. The actual task and true challenge is to gain insight into the structure of members of proper minor-closed classes of graphs or matroids. The well-quasi-ordering and minor-testing results are consequences – not necessarily easy ones – of the structure that is uncovered. Ironically, while one may begin studying structure with the purpose of obtaining marketable results, in the end it is probably the structural theorems themselves that are the most satisfying aspect of a project like this. To acquire that structural insight is the bulk of the work and the theorems that in the end describe the entire structure are the main deliveries of a project like this.

Introduction

The triangle removal lemma states that for every ε > 0 there exists δ > 0 such that any graph on n vertices with at most δn3 triangles may be made triangle-free by removing at most εn2 edges. This result, proved by Ruzsa and Szemerédi [94] in 1976, was originally stated in rather different language.

The original formulation was in terms of the (6, 3)-problem. This asks for the maximum number of edges f(3)(n, 6, 3) in a 3-uniform hypergraph on n vertices such that no 6 vertices contain 3 edges. Answering a question of Brown, Erdὄs and Sós [19], Ruzsa and Szemerédi showed that f(3)(n, 6, 3) = o(n2). Their proof used several iterations of an early version of Szemerédi's regularity lemma [111].

This result, developed by Szemerédi in his proof of the Erdὄos-Turán conjecture on arithmetic progressions in dense sets [110], states that every graph may be partitioned into a small number of vertex sets so that the graph between almost every pair of vertex sets is random-like.