Existence of Configurations
Generally speaking, Combinatorics deals with two different kinds of problems: those in which we want to count the number of distinct ways of making a certain choice and those in which we want to make sure that some configuration does appear. In order to solve a problem of the first kind above, we employ, among others, the counting techniques discussed in the previous chapters. For the second kind of problem, up to this moment we have not developed any idea that could be systematically used. It is our purpose in this chapter to remedy this state of things. To this end, we start by discussing the famous pigeonhole’s principle of Dirichlet, along with several interesting examples. Then we move on to some applications of the principle of mathematical induction to the existence of configurations. The chapter continues with the study of partial order relations, exploring Mirsky’s theorem on the relation between chains and anti-chains. We close the chapter by explaining how, in some situations, the search for an adequate invariant or a semi-invariant can give the final outcome of certain seemingly random algorithms.