Simple extensions of combinatorial structures

TL;DR

This paper investigates how arbitrary combinatorial structures can be embedded into simple structures within various classes, providing tight bounds on the minimal number of elements needed for such embeddings.

Contribution

It establishes optimal bounds on the number of elements required to embed any structure into a simple one across multiple combinatorial classes.

Findings

Embedding bounds are tight for all classes considered.

The minimal number of added elements varies with the class, from 1 to logarithmic functions.

Results unify and extend previous understanding of simple structures in combinatorics.

Abstract

An interval in a combinatorial structure S is a set I of points which relate to every point from S 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 2. The function f(n) in these cases is 2, \lceil log_2(n+1)\rceil, \lceil (n+1)/2\rceil, \lceil (n+1)/2\rceil, \lceil log_4(n+1)\rceil, \lceil \log_3(n+1)\rceil and 1, respectively. In each case these bounds are best possible.