Strong Partition Relations Below the Power Set: Consistency, Was Sierpi\'nski Right, II?

by Shelah. [Sh:288]
Proc Conference on Set Theory and its Applications in honor of A.Hajnal and V.T.Sos, Budapest, 1/91, 1991
We continue here [Sh276] but we do not relay 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 Erdos 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 .

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