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increasing mapping on set power

Source: 16-th Hungary-Israel Binational Mathematical Competition 2005

March 29, 2007
combinatorics unsolvedcombinatorics

Problem Statement

Let ff be an increasing mapping from the family of subsets of a given finite set HH into itself, i.e. such that for every X⊆Y⊆HX \subseteq Y\subseteq H we have f(X)⊆f(Y)⊆H.f (X )\subseteq f (Y )\subseteq H . Prove that there exists a subset H0H_{0} of HH such that f(H0)=H0.f (H_{0}) = H_{0}.
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