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Combinatorial theorems on contractive mappings in power sets

โœ Scribed by Egbert Harzheim

Publisher
Elsevier Science
Year
1982
Tongue
English
Weight
713 KB
Volume
40
Category
Article
ISSN
0012-365X

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โœฆ Synopsis

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 f a mapping of the pow+:r set 13 of S into b such that f(T) E T for all TE '@, then there exists a strictly increasing sequence TI c ---c T,, oi' subsets of S such that one of the following three :~ssibilities holds: (a) all sets flITi), i = 1 ,.. .,n,are equal; (b) for all i=l,..., n, we have f(Ti) = Ti; (c) F =f(Ti+l) for all i = 1 ,-*.I n-l. This theorem generalizes theorems of the author, Rado, and Leeb. It has applications for subtrees in power sets.

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