On a question of Andreas Weiermann

The goal of this paper is to give some information about the following question, posed by Andreas Weiermann (private communication). Is it true that for every β, γ < ε0 there exist α so that whenever A is α–large, A satisfies some inessential assumption, say min(A) ≥ 2, and G : A → β is such that ∀a ∈ A psn(G(a)) ≤ a there must exist γ–large C ⊆ A on which G is nondecreasing. Here ε0 is the smallest ordinal solution to the equation ω = ε, the notion of α–largeness is in the sense of the so–called Hardy hierarchy and psn(α) is the greatest natural number which occurs in the full Cantor normal expansion of α. We answer this positively. We derive this result from a partition theorem of J. Ketonen and R. Solovay [10] as reworked for the Hardy hierarchy in [1, 2, 3] and [12], in particular from theorem 5 in [2]. Later we obtain much sharper upper bound for α in terms of β and γ for very small ordinals β. Now, we write something on the motivations which stand behind this work. One of the open problems of proof theory is to determine the exact strength of the Ramsey’s theorem for pairs (RT2) stated as the sentence of second order arithmetic