What is the largest number of subsets of an n-element set that can be selected, so that any two of these subsets have at least one element in common? What is the largest number of terms of a sequence of real numbers that does not contain a monotone subsequence with n terms? Questions of this type may be answered by appealing to some simple but extremely useful principles, such as the pigeonhole principle, and by studying fundamental concepts and structures of discrete mathematics, such as set partitions (or equivalence relations) and partial orders. This chapter serves as a brief introduction to these topics.