Kuratowski's free set theorem explained
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 \Phi\colon[X]n\to[X]<\omega
, 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, Kuratowski's free set theorem is superseded by Hajnal's set mapping theorem.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.
