# Combinatorial Order Theory

• Martin Aigner
Chapter
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 234)

## Abstract

Under the general term “combinatorial order theory” we want to collect some results on posets by concentrating less on the structure of posets than on properties present in any poset, such as chains, antichains, matchings, etc. Typical problems to be considered are the determination of the minimal number of chains into which a finite poset can be decomposed or the existence of a matching between the points and copoints of a ranked poset. In fact, the importance of this branch of combinatorial mathematics derives to a large extent from the fact that most of the main results are existence theorems supplementing the many counting results established in chapters III to V. To testify to the broad range of applications, we have included a variety of examples from different sources (graphs, networks, 0,1-matrices, etc.). Each of the sections is headed by a basic theorem after which we study variations of the main theorem, applications, and the mutual dependence with other results.

## Keywords

Bipartite Graph Rank Function Maximal Chain Ramsey Number Transversal Theorem
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