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| '''Kuratowski's free set theorem''', named after [[Kazimierz Kuratowski]], is a result of [[set theory]], an area of [[mathematics]]. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several [[lattice theory]] problems, such as the [[Congruence Lattice Problem]].
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| Denote by <math>[X]^{<\omega}</math> the [[Set (mathematics)|set]] of all [[Finite set|finite subsets]] of a set <math>X</math>. Likewise, for a [[positive integer]] <math>n</math>, denote by <math>[X]^n</math> the set of all <math>n</math>-elements subsets of <math>X</math>. For a [[Map (mathematics)|mapping]] <math>\Phi\colon[X]^n\to[X]^{<\omega}</math>, we say that a [[subset]] <math>U</math> of <math>X</math> is ''free'' (with respect to <math>\Phi</math>), if <math>u\notin\Phi(V)</math>, for any <math>n</math>-element subset <math>V</math> of <math>U</math> and any <math>u\in U\setminus V</math>. [[Kuratowski]] published in 1951 the following result, which characterizes the [[Infinity|infinite]] [[Cardinal number|cardinals]] of the form <math>\aleph_n</math>.
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| The theorem states the following. Let <math>n</math> be a positive integer and let <math>X</math> be a set. Then the [[cardinality]] of <math>X</math> is greater than or equal to <math>\aleph_n</math> if and only if for every mapping <math>\Phi</math> from <math>[X]^n</math> to <math>[X]^{<\omega}</math>,
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| there exists an <math>(n+1)</math>-element free subset of <math>X</math> with respect to <math>\Phi</math>.
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| For <math>n=1</math>, Kuratowski's free set theorem is superseded by [[Hajnal's set mapping theorem]].
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| == References ==
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| * [[Paul Erdős|P. Erdős]], [[András Hajnal|A. Hajnal]], A. Máté, [[Richard Rado|R. Rado]]: ''Combinatorial Set Theory: Partition Relations for Cardinals'', North-Holland, 1984, pp. 282-285.
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| * [[Kazimierz Kuratowski|C. Kuratowski]], ''Sur une caractérisation des alephs'', Fund. Math. '''38''' (1951), 14--17.
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| * John C. Simms: Sierpiński's theorem, ''Simon Stevin'', '''65''' (1991) 69--163.
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| {{settheory-stub}}
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| [[Category:Set theory]]
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