A partition theorem for a large dense linear order

Abstract

Let κ be a cardinal which is measurable after generically adding ℶ_κ+ω many Cohen subsets to κ, and let ℚ_κ = (Q, ≤ _Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t ⁺ _m such that the set [Q]^m of m-tuples from Q can be partitioned into classes 〈C _i : i < t ⁺ _m }〉 such that for any coloring a class C _i in fewer than κ colors, there is a copy ℚ* of ℚ_κ such that [ℚ*]^m ⋂ C _i is monochromatic. It follows that $\mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m$, that is, for any coloring of [ℚ_κ]^m with fewer than κ colors there is a copy Q′ ⊆ Q of ℚ_κ such that [Q′]^m has at most t ⁺ _m colors. On the other hand, we show that there are colorings of ℚ_κ such that if Q′ ⊆ Q is any copy of ℚ_κ then C _i ⋂ [Q′] ≠ ø; for all i < t ⁺ _m , and hence $\mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m$.

We characterize t ⁺ _m as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t ⁺₂ = 2 and for m > 2 we have t ⁺ _m > t _m , the m-th tangent number.

The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ^+ω-strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah’s theorem, is not directly implied by any large cardinal assumption.