We present two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ, ω + 1), although λ is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, [7], that, for example, consistently, אω1 6→ (אω1 , ω+1) 2 and consistently, 20 6→ (20 , ω + 1).