Cohen Sets and Consistent Extensions of the Erdös - Dushnik - Miller

We present two different types of models where, for certain singular cardinals λ of uncountable cofinality, λ → (λ, ω + 1) 2 , although λ is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, [7], that, for example, consistently, ℵω 1 → (ℵω 1 , ω + 1) 2 and consistently, 2 ℵ 0 → (2 ℵ 0 , ω + 1) 2. §0. INTRODUCTION. For regular uncountable κ, the Erdös-Dushnik-Miller theorem, Theorem 11.3 of [2], states that κ → (κ, ω + 1) 2. For singular cardinals, κ, they were only able to obtain the weaker result, Theorem 11.1 of [1], that κ → (κ, ω) 2. It is not hard to see that if cf κ = ω then κ → (κ, ω + 1) 2. If cf κ > ω and κ is a strong limit cardinal, then it follows from the General Canonization Lemma, Lemma 28.1 of [1], that κ → (κ, ω + 1) 2. Question 11.4 of [1] is whether this holds without the assumption that κ is a strong limit cardinal, (1) ℵ ω1 → (ℵ ω1 , ω + 1) 2. Another natural question, which the second author first heard from Todorcevic, is whether, in ZFC, (2) 2 ℵ0 → (2 ℵ0 , ω + 1) 2. In connection with (2), we note that the first author proved, [2], §2, the consistency of 2 ℵ0 → [ℵ 1 ] 2 n,2. In this paper we address these questions, by presenting two types of models where there is a singular cardinal λ of uncountable cofinality, such that λ → (λ, ω + 1) 2 even though λ is not a strong limit cardinal. In either model, λ can be taken to be ℵ ω1 and in the second, we can also have, simultaneously, λ = 2 ℵ0. We also announce here, and will present in a subsequent paper, some very recent results that show that, consistently, (1) and (2) above may fail. For (1), this answers Question 11.4 of [1] negatively. The first type of model seems specific to having the order type of the homogeneous set for the second color (green, for us, whereas the first color is the " traditional " red) be ω + 1, whereas the second model allows generalizations to green homogeneous sets of order type θ + 1 for …