 The first of these theorems says that if \(n,r\) are finite positive numbers and \(X\) is an infinite set and we partition the set of subsets of \(X\) of size \(n\) into \(r\) parts then there is an infinite subset \(Y\) of \(X\) such that every subset of \(Y\) of size \(n\) is in the same part. The second theorem gives a version of the first, but for finite sets of sufficiently large size. It says that, for any finite positive numbers \(m,n,r\) there is a finite positive \(k\) such that if \(X\) has at least \(k\) elements, and we partition the sets of subsets of \(X\) of size \(n\) into \(r\) parts, then there is a subset \(Y\) of \(X\) with at least \(m\) elements such that every subset of \(Y\) of size \(n\) is in the same part of the partition. Ramsey gave an upper bound on such a \(k\) in terms of \(m,n,r\) but even today the actual value of the least such \(k\), known as the Ramsey Number \(R_n(m,r)\), is unknown in all but a handful of cases for small \(n,m,r\). These two results are now referred to as the Infinite and Finite Ramsey Theorems, respectively. They can be viewed as generalizing Dirichlet’s much earlier Schubfachprinzip or Pigeon Hole Principle (1832).