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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.

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For n=1, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.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.

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