### 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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