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

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

For n=1, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.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: Sierpiński's theorem, Simon Stevin, 65 (1991) 69--163.