Kuratowski's free set theorem

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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 for any n-element subset V of U and any u\in U\setminus V, u\notin\Phi(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For n=1, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.


  • 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.