A Variation on a Theorem of Galvin and Hajnal

Let ℵ η be a singular cardinal of regular uncountable cofinality κ. Let {η(ξ): ξ < κ} be a continuous increasing sequence with limit η, and let κ ξ =ℵ η(ξ)+Ф(ξ) ,ξ < κ be regular cardinals. Let I be a normal ideal on κ, and assume that the reduced product ∏ ξ<κ κ ξ / I admits a cofinal λ‐scale of ordinal functions. Then λ ⩽ ℵ η+τ , where τ=∥Ф∥ I is the I ‐norm of Ф.