Combinatorial theorems on contractive mappings in power sets

We prove that to every positive integer n there exists a positive integer h such that the following holds: If S is a set of h elements and ? a mapping of the power set of S into such that ?(T)T for all T , then there exists a strictly increasing sequence T₁T_n of subsets of S such that one of the following three possibilities holds: (a) all sets ?(T_i), i= 1,…,n, are equal; (b) for all i=1,…, n, we have ?(T_i)=T_i; (c) T_i=?(T_i+1) for all i= 1,…,n-1. This theorem generalizes theorems of the author, Rado, and Leeb. It has applications for subtrees in power sets.