Negative partition relations for ultrafilters on uncountable cardinals

Abstract

Assuming GCH, we prove that if κ \kappa is a successor cardinal and U U is a uniform ultrafilter on κ \kappa , then U ↛ ( U , 3 ) 2 U \nrightarrow {(U,3)^2} . The case κ = ω 1 \kappa = {\omega _1} is an old result of Hajnal. Our proof makes use of several known results concerning nonregular, weakly normal and indecomposable ultrafilters, as well as some negative partition relations for uncountable ordinals.