Sh:909

• Gruenhut, E., & Shelah, S. (2011). Uniforming n-place functions on well founded trees. In Set theory and its applications, Vol. 533, Amer. Math. Soc., Providence, RI, pp. 267–280. arXiv: 0906.3055 DOI: 10.1090/conm/533/10512 MR: 2777753
• Abstract:
  In this paper the Erdős-Rado theorem is generalized to the class of well founded trees. We define an equivalence relation on the class {\rm rs}(\infty)^{<\aleph_0} (finite sequences of decreasing sequences of ordinals) with \aleph_0 equivalence classes, and for n<\omega a notion of n-end-uniformity for a colouring of {\rm rs}(\infty)^{<\aleph_0} with \mu colours. We then show that for every ordinal \alpha, n<\omega and cardinal \mu there is an ordinal \lambda so that for any colouring c of T={\rm rs}(\lambda)^{<\aleph_0} with \mu colours, T contains S isomorphic to {\rm rs}(\alpha) so that c\restriction S^{<\aleph_0} is n-end uniform. For c with domain T^n this is equivalent to finding S\subseteq T isomorphic to {\rm rs}(\alpha) so that c\upharpoonright S^{n} depends only on the equivalence class of the defined relation, so in particular T\rightarrow({\rm rs}(\alpha))^n_{\mu,\aleph_0}. We also draw a conclusion on colourings of n-tuples from a scattered linear order.
• Version 2009-10-25_11 (14p) published version (14p)

@incollection{Sh:909,
 author = {Gruenhut, Esther and Shelah, Saharon},
 title = {{Uniforming $n$-place functions on well founded trees}},
 booktitle = {{Set theory and its applications}},
 series = {Contemp. Math.},
 volume = {533},
 year = {2011},
 pages = {267--280},
 publisher = {Amer. Math. Soc., Providence, RI},
 mrnumber = {2777753},
 mrclass = {03E02 (05D10)},
 doi = {10.1090/conm/533/10512},
 note = {\href{https://arxiv.org/abs/0906.3055}{arXiv: 0906.3055}},
 arxiv_number = {0906.3055}
}