Strong Partition Realations Below the Power Set : Consistency Was Sierpinski Right ? II

We continue here [Sh276] (see the introduction there) but we do not relay on it. The motivation was a conjecture of Galvin stating that 2 ω ≥ ω 2 + ω 2 → [ω 1 ] n h(n) is consistent for a suitable h : ω → ω. In section 5 we disprove this and give similar negative results. In section 3 we prove the consistency of the conjecture replacing ω 2 by 2 ω , which is quite large, starting with an Erd˝ os cardinal. In section 1 we present iteration lemmas which needs when we replace ω by a larger λ and in section 4 we generalize a theorem of Halpern and Lauchli replacing ω by a larger λ.