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

< ω

the set of all finite subsets of a set

Likewise, for a positive integer

, denote by

, we say that a subset

is free (with respect to

-element subset

Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form

The theorem states the following.

be a positive integer and let

Then the cardinality of

is greater than or equal to

-element free subset of

, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.

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