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For $n=1$, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.

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 (1991) "Sierpiński's theorem", Simon Stevin 65: 69–163.