This paper proves that every isolated set A satisfies the following generalized pigeonhole principle: if g: \(A\to B\) and \([A]>n[B]\), then \([g^{-1}(b)]>n\) for some \(b\in B\). ([A] denotes the recursive equivalence type of A.) However, the analogous principle, using an arbitrary isol in place of the number n, will not hold in general for isolated A - although it will hold if [A] is multiple-free. Other related notions and applications are also treated.