FOM: Unscientific survey

[JoeShipman@aol.com]

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