Combinatorial theorems on contractive mappings in power sets

Abstract

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 ƒ a mapping of the power set B of S into B such that ƒ( T)⊆ T for all T∈ B , then there exists a strictly increasing sequence T 1∋⋰∋ T n of subsets of S such that one of the following three possibilities holds: (a) all sets ƒ(T i) , i= 1,…, n, are equal; (b) for all i=1,…, n, we have ƒ(T i)=T i; (c) T i=ƒ(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.