  We continue here [She88] but we do not rely on it. The motivation was a
conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2->
[omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section
5 we disprove this and give similar negative results. In section 3 we prove the
consistency of the conjecture replacing omega_2 by 2^omega, which is quite
large, starting with an Erd\H{o}s cardinal. In section 1 we present iteration
lemmas which are needed when we replace omega by a larger lambda and in section
4 we generalize a theorem of Halpern and Lauchli replacing omega by a larger
lambda .
