  Let Q_K=(Q,<_Q)$ be a strongly K-dense linear order of size K for a suitable
cardinal K. We prove, for all integers m > 1 that there is a finite value t_m^+
such that the set of all m-tuples from Q can be divided into t_m^+ many
classes, such that whenever any of these classes C is colored with fewer than K
many colors, there is a copy Q* of Q_K such that all m-tuples from Q* in C
receive the same color. As a consequence we obtain that whenever we color the
m-tuples of Q with fewer than K many colors, there is a copy of Q_K all
m-tuples from which are colored in at most t_m^+ colors. In other words, the
partition relation Q_K -->(Q_K)^m_{<K,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 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 m-th tangent
number. The paper also contains similar partition results about K-Rado graphs.
A consequence of our work and some earlier results of Hajnal and Komjath 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.
