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SIMPLE EXTENSIONS OF COMBINATORIAL STRUCTURES

Part of: Model theory Ordered sets Combinatorics Graph theory

Published online by Cambridge University Press:  21 December 2010

Robert Brignall ,
Nik Ruškuc  and
Vincent Vatter
Robert Brignall
Affiliation:
Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, England (email: r.brignall@open.ac.uk)
Nik Ruškuc
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, Scotland (email: nik@mcs.st-and.ac.uk)
Vincent Vatter
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, U.S.A. (email: vatter@ufl.edu)
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Abstract

An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.

MSC classification

Secondary: 05C20: Directed graphs (digraphs), tournaments 06A06: Partial order, general 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 05A05: Permutations, words, matrices 03C13: Finite structures

Information

Type
Research Article
Information
Mathematika , Volume 57 , Issue 2 , July 2011 , pp. 193 - 214
Copyright
Copyright © University College London 2010

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