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1 - Hereditary and monotone properties of combinatorial structures

Published online by Cambridge University Press:  16 March 2010

Anthony Hilton
Affiliation:
University of Reading
John Talbot
Affiliation:
University College London
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Summary

Abstract

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 ↦ |Pn| the labelled speed of P. Similarly, the unlabelled speed is n ↦ |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.

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Publisher: Cambridge University Press
Print publication year: 2007

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