# Sh:288

- Shelah, S. (1992).
*Strong partition relations below the power set: consistency; was Sierpiński right? II*. In Sets, graphs and numbers (Budapest, 1991), Vol. 60, North-Holland, Amsterdam, pp. 637–668. arXiv: math/9201244 MR: 1218224 -
Abstract:

We continue here [Sh:276] (see the introduction there) but we do not rely on it. The motivation was a conjecture of Galvin stating that \big( 2^{\omega}\ge \omega_2 \big) + \big( \omega_2\to [\omega_1]^{n}_{h(n)} \big) is consistent for a suitable h : \omega \to \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 Erdős 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.This is a slightly corrected version of an old work.

- Version 2024-01-29 (24p) published version (32p)

Bib entry

@incollection{Sh:288, author = {Shelah, Saharon}, title = {{Strong partition relations below the power set: consistency; was Sierpi\'nski right? II}}, booktitle = {{Sets, graphs and numbers (Budapest, 1991)}}, series = {Colloq. Math. Soc. J\'anos Bolyai}, volume = {60}, year = {1992}, pages = {637--668}, publisher = {North-Holland, Amsterdam}, mrnumber = {1218224}, mrclass = {03E05 (03E35)}, note = {\href{https://arxiv.org/abs/math/9201244}{arXiv: math/9201244}}, arxiv_number = {math/9201244} }