un 2 00 5 A partition theorem for a large dense linear order

Let Q κ = (Q, ≤ Q) be a strongly κ-dense linear order of size κ for κ a suitable cardinal. 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 divided into t + m many classes, such that whenever any of these classes C is colored with < κ many colors, there is a copy Q * of Q κ such that [Q * ] m ∩ C is monochromatic. As a consequence we obtain that whenever we color [Q κ ] m with < κ many colors, there is a copy of Q κ all m-tuples from * Support by EPSRC through an Advanced Fellowship is gratefully acknowledged. † Support by EPSRC and the University of East Anglia during the period when the project was started, and by the University of Münster, during the writing of the first draft of the paper is gratefully acknowledged. 1 which are colored in at most t + m colors. In other words, the partition relation Q κ → (Q κ) m <κ/r holds for some finite r = t + m. We show that t + m is the minimal value with this property. We were not able to give a formula for t + m but we can describe t + m as the cardinality of a certain finite set of types. We also give an upper and a lower bound on its value and for m = 2 we obtain t + 2 = 2, while for m > 2 we have t + m > t m , the mth tangent number. The paper also contains similar positive partition results about κ-Rado graphs. A consequence of our work and some earlier results of Hajnal and Komjáth is that a theorem of Shelah known to follow from a large cardinal assumption in a generic extension, does not follow from any large cardinal assumption on its own.