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 ${\displaystyle [X]^{<\omega }}$ the set of all finite subsets of a set ${\displaystyle X}$. Likewise, for a positive integer ${\displaystyle n}$, denote by ${\displaystyle [X]^{n}}$ the set of all ${\displaystyle n}$-elements subsets of ${\displaystyle X}$. For a mapping ${\displaystyle \Phi \colon [X]^{n}\to [X]^{<\omega }}$, we say that a subset ${\displaystyle U}$ of ${\displaystyle X}$ is free (with respect to ${\displaystyle \Phi }$), if for any ${\displaystyle n}$-element subset ${\displaystyle V}$ of ${\displaystyle U}$ and any ${\displaystyle u\in U\setminus V}$, ${\displaystyle u\notin \Phi (V)}$. Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form ${\displaystyle \aleph _{n}}$.

The theorem states the following. Let ${\displaystyle n}$ be a positive integer and let ${\displaystyle X}$ be a set. Then the cardinality of ${\displaystyle X}$ is greater than or equal to ${\displaystyle \aleph _{n}}$ if and only if for every mapping ${\displaystyle \Phi }$ from ${\displaystyle [X]^{n}}$ to ${\displaystyle [X]^{<\omega }}$, there exists an ${\displaystyle (n+1)}$-element free subset of ${\displaystyle X}$ with respect to ${\displaystyle \Phi }$.

For ${\displaystyle n=1}$, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.

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