# Kuratowski's free set theorem

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For ${\displaystyle n=1}$, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.