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
]
<
ω
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
Φ
:
[
X
]
n
→
[
X
]
<
ω
, we say that a subset
U
of
X
is free (with respect to
Φ
), if for any
n
-element subset
V
of
U
and any
u
∈
U
∖
V
,
u
∉
Φ
(
V
)
,. Kuratowski published in 1951 the following result, which characterizes the infinite cardinals of the form
ℵ
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
ℵ
n
if and only if for every mapping
Φ
from
[
X
]
n
to
[
X
]
<
ω
, there exists an
(
n
+
1
)
-element free subset of
X
with respect to
Φ
.
For
n
=
1
, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.
