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Combinatorial theorems on contractive mappings in power sets
No full textAbstractWe 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 T1∋⋰∋Tn of subsets of S such that one of the following three possibilities holds: (a) all sets ƒ(Ti), i= 1,…,n, are equal; (b) for all i=1,…, n, we have ƒ(Ti)=Ti; (c) Ti=ƒ(Ti+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
