A partition theorem for a large dense linear order. (English summary) Israel J. Math. 171 (2009), 237–284. Denote by ≤Q the natural κ-dense linear order on κ> 2 which extends the lexicographic order. Write Qκ = 〈 κ> 2, ≤Q〉. For regular κ the linear order Qκ is κ-dense. For κ = ω, Q = Qω has the order type of the rationals. Devlin proved that Q → (Q) n <ω,tn and Q → (Q)n <ω,tn−1, where the tn is the nth tangent number, i.e., tan(x) = ∑ ∞ n x2n−1 tn (2n−1)!. The main result of the paper is the following theorem: For every natural number m there is t + m ∈ ω such that for any cardinal κ which is measurable after generically adding many Cohen subsets of κ we have Qκ → (Qκ) n <ω,t + n and Qκ → (Qκ) n <ω,t + n −1. The authors characterize t + m as the cardinality of a certain finite set of ordered trees. In particular
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