Kuratowski's free set theorem

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. Denote by [math]\displaystyle{ [X]^{\lt \omega} }[/math] the set of all finite subsets of a set [math]\displaystyle{ X }[/math]. Likewise, for a positive integer [math]\displaystyle{ n }[/math], denote by [math]\displaystyle{ [X]^n }[/math] the set of all [math]\displaystyle{ n }[/math]-elements subsets of [math]\displaystyle{ X }[/math]. For a mapping [math]\displaystyle{ \Phi\colon[X]^n\to[X]^{\lt \omega} }[/math], we say that a subset [math]\displaystyle{ U }[/math] of [math]\displaystyle{ X }[/math] is free (with respect to [math]\displaystyle{ \Phi }[/math]), if for any [math]\displaystyle{ n }[/math]-element subset [math]\displaystyle{ V }[/math] of [math]\displaystyle{ U }[/math] and any [math]\displaystyle{ u\in U\setminus V }[/math], [math]\displaystyle{ u\notin\Phi(V) }[/math]. Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form [math]\displaystyle{ \aleph_n }[/math].

The theorem states the following. Let [math]\displaystyle{ n }[/math] be a positive integer and let [math]\displaystyle{ X }[/math] be a set. Then the cardinality of [math]\displaystyle{ X }[/math] is greater than or equal to [math]\displaystyle{ \aleph_n }[/math] if and only if for every mapping [math]\displaystyle{ \Phi }[/math] from [math]\displaystyle{ [X]^n }[/math] to [math]\displaystyle{ [X]^{\lt \omega} }[/math], there exists an [math]\displaystyle{ (n+1) }[/math]-element free subset of [math]\displaystyle{ X }[/math] with respect to [math]\displaystyle{ \Phi }[/math].

For [math]\displaystyle{ n=1 }[/math], Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.

References

• P. Erdős, A. Hajnal, A. Máté, R. Rado: Combinatorial Set Theory: Partition Relations for Cardinals, North-Holland, 1984, pp. 282–285.
• C. Kuratowski, Sur une caractérisation des alephs, Fund. Math. 38 (1951), 14–17.
• John C. Simms (1991) "Sierpiński's theorem", Simon Stevin 65: 69–163.