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.

Denote by $[X] ^{the set of all finite subsets of a set X . Likewise, for a positive integer n, denote by [X] ^n the set of all n -elements subsets of X . For a mapping Phicolon [X] ^n o [X] ^{, we say that a subset U of X is "free" (with respect to Phi), if u
otinPhi(V), for any n -element subset V of U and any uin Usetminus V . Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form aleph_n .$

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 $[X] ^{,there exists an (n+1) -element free subset of X with respect to Phi .$

For $n=1$ , Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.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

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