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)

@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}
}