Coloring Theorems at successors of singular cardinals
by
Todd Eisworth
University of Northern Iowa
Coauthors: Saharon Shelah (Hebrew University of Jerusalem)

One of the earliest applications of pcf theory was Shelah's proof that \aleph_\omega+1 is not a Jonsson cardinal. [\kappa is a Jonsson cardinal if for some "large enough" regular cardinal \chi, there is an elementary submodel M of H(\chi) such that \kappa in M, |M \cap \kappa|=\kappa, but \kappa is not a subset of M; this is known to be equivalent to the negated partition relation \kappa\nrightarrow [\kappa] ^{< \omega}.] The question of whether the successor of a singular cardinal can be Jonsson is still open; pcf theory and club-guessing put many restrictions on potential models of this phenomenon. The work we will present shows that in many situations, the successor \lambda of a singular cardinal will fail to be Jonsson in a spectacular way - we will be able to find a coloring of pairs of ordinals less that \lambda that exhibits "complicated" behavior on any subset of \lambda of size \lambda. The results have applications in topology - for example, they say something about chain conditions, generalized S-spaces, and generalized L-spaces.

Date received: February 22, 2002