Three famous theorems on finite sets

Abstract

In this chapter we are concerned with a basic theme of combinatorics: properties and sizes of special families $$\mathcal{F}$$of subsets of a finite set N = {1, 2, . . . , n}. We start with two results which are classics in the field: the theorems of Sperner and of Erdős–Ko–Rado. These two results have in common that they were reproved many times and that each of them initiated a new field of combinatorial set theory. For both theorems, induction seems to be the natural method, but the arguments we are going to discuss are quite different and truly inspired.