Partition relations for successor cardinals

Abstract

This paper investigates the relations κ + → ( α) 2 2 and its variants for uncountable cardinals κ. First of all, the extensive literature in this area is reviewed. Then, some possibilities afforded by large cardinal hypotheses are derived, for example, if κ is measurable, then κ + → ( κ + κ + 1, α) 2 2 for every α < κ +. Finally, the limitations imposed on provability in ZFC by L and by relative consistency via forcing are considered, primarily the consistency of: if κ is not weakly compact, then κ + ⇅ [κ : κ] 2 κ .