A partition theorem for a large dense linear order

Abstract

Let κ be a cardinal which is measurable after generically adding ℶκ+ω many Cohen subsets to κ, and let ℚκ = (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 + such that the set [Q] m of m-tuples from Q can be partitioned into classes 〈C i : i < t + }〉 such that for any coloring a class C i in fewer than κ colors, there is a copy ℚ* of ℚκ 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 [ℚκ] m with fewer than κ colors there is a copy Q′ ⊆ Q of ℚκ such that [Q′] m has at most t + colors. On the other hand, we show that there are colorings of ℚκ such that if Q′ ⊆ Q is any copy of ℚκ then C i ⋂ [Q′] ≠ ø; for all i < t + , and hence $$ \mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m $$ . We characterize t + 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 + > 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.