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For $$n=1$$, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.