SIMPLE EXTENSIONS OF COMBINATORIAL STRUCTURES

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 can be embedded into a simple structure from by adding at most f(n) elements. We prove such results when 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.