21 st British Combinatorial Conference University of Reading 9 th - 13 th July 2007

A hereditary property of graphs is a collection of (isomorphism classes of) graphs which is closed under taking induced graphs, and contains arbitrarily large structures. Given a family F of graphs, the family P(F) of graphs containing no member of F as an induced subgraph is a hereditary property, and every hereditary property of graphs arises in this way. A hereditary property of other combinatorial structures is defined analogously. A property is monotone if it is closed under taking (not necessarily induced) substructures. Given a property P, we write Pn for the number of distinct structures with vertices labelled 1, . . . , n, and call the function n 7→ |Pn| the labelled speed of P. Similarly, the unlabelled speed is n 7→ |Pn|, where Pn is the set of distinct structures with n unlabelled vertices. The study of hereditary properties is on the borderline of extremal, enumerative, and probabilistic combinatorics. Thus, for a family F of graphs, the problem of determining the speed of P(F) is a natural extension of the basic question in extremal graph theory concerned with the maximal number of edges in a graph of order n containing no member of F as a subgraph. For many a combinatorial structure (graphs, posets, partitions, words, etc.), there is a surprising phase transition: the speed jumps from one range to a much higher one. Thus the speed of a property is either not much larger than a certain function f(n) or is at least as large as a function F which is much larger than f . Although the jumps may look fairly similar for a variety of combinatorial structures, much of the time their proofs need new ideas, and give deep insights into the structures. In the past few decades, much research has been done on hereditary and monotone properties of a number of combinatorial structures: the aim of this paper is to review some of these results, with special emphasis on the most recent results.