Topics in Combinatorics

This is a follow-up to the handbook Introduction to Combinatorics, written by the same two authors, which was published in 2015. The topics covered here require more mathematical preparation than those in the earlier volume, but the style is deliberately discursive, with the explicit aim of exploring how to go about solving challenging problems, rather than just describing the finished solution. The book begins with a number of classical problems including Lucas’s problème des ménages, which counts arrangements of guests around a table, Hall’s marriage theorem, which concerns the effective allocation of resources, and Kirkman’s schoolgirl problem, which was published in a recreational puzzle magazine for Victorian ladies. It then explores major areas of the subject including graph theory, non-partisan games, Pólya enumeration and partitions, finishing with a chapter on Ramsey theory, which can be described as ‘finding order in disorder’. The book concludes with a result which allows readers to prove something which is, in a clearly defined sense, unprovable. There are dozens of carefully chosen exercises and, as before, there are full solutions.