FOM: Unscientific survey
JoeShipman@aol.com
JoeShipman at aol.com
Thu May 31 16:11:11 EDT 2001
In my opinion, Ramsey's theorem is not a theorem in "Cardinal Arithmetic".
However, there is a standard relation between quadruples of cardinals
A(k,l,m,n) which holds iff whenever the l-subsets of a set of cardinality n
are k-colored there exists a monochromatic subset of size m (a subset of size
m all of whose l-subsets have the same color). (This can also be though of
as a relation betweem two cardinals k,l and two ordinals m,n where you ask
for a subset of n of order type m rather than cardinality m, but the
restricted version using only cardinals is of more general interest.)
The classical Ramsey's theorem states A(k,l,m,n) for k and l any finite
numbers and m and n countably infinite. This 4-ary relation between
cardinals is the first topic in combinatorial set theory AFTER cardinal
arithmetic, but properly speaking, "cardinal arithmetic" is concerned only
with the cardinal functions of addition, multiplication, exponentiation, and
cofinality.
-- Joe Shipman
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