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  the set of all finite subsets of a set . Likewise, for a positive integer , denote by  the set of all -elements subsets of . For a mapping , we say that a subset  of  is free (with respect to ), if for any -element subset  of  and any , . Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form . The theorem states the following. Let  be a positive integer and let  be a set. Then the cardinality of  is greater than or equal to  if and only if for every mapping  from  to ,
there exists an -element free subset of  with respect to . For , Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.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. Set theory