Ordinal Partitions

Lawrence C. Paulson 🌐

August 3, 2020

Abstract

The theory of partition relations concerns generalisations of Ramsey's theorem. For any ordinal α, write α(α,m)2 if for each function f from unordered pairs of elements of α into {0,1}, either there is a subset Xα order-isomorphic to α such that f{x,y}=0 for all {x,y}X, or there is an m element set Yα such that f{x,y}=1 for all {x,y}Y. (In both cases, with {x,y} we require xy.) In particular, the infinite Ramsey theorem can be written in this notation as ω(ω,ω)2, or if we restrict m to the positive integers as above, then ω(ω,m)2 for all m. This entry formalises Larson's proof of ωω(ωω,m)2 along with a similar proof of a result due to Specker: ω2(ω2,m)2. Also proved is a necessary result by Erdős and Milner: ω1+αn(ω1+α,2n)2.

License

BSD License

Topics

Session Ordinal_Partitions