Midpoint circle algorithm

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 $\mathrm{\Phi }:[X{]}^n\to [X{]}^{<\omega }$, we say that a subset $U$ of $X$ is free (with respect to $\mathrm{\Phi }$), if $u\notin \mathrm{\Phi }(V)$, for any $n$-element subset $V$ of $U$ and any $u\in U\setminus V$. Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form ${\mathrm{\aleph }}_n$.

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 ${\mathrm{\aleph }}_n$ if and only if for every mapping $\mathrm{\Phi }$ from $[X{]}^n$ to $[X{]}^{<\omega }$, there exists an $(n+1)$-element free subset of $X$ with respect to $\mathrm{\Phi }$.

For $n=1$, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.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: Sierpiński's theorem, Simon Stevin, 65 (1991) 69--163.