# Sh:679

• Shelah, S. (2002). A partition theorem. Sci. Math. Jpn., 56(2), 413–438.
• Abstract:
We prove the following: there is a primitive recursive function f_{-}^*(-,-), in the three variables, such that: for every natural numbers t,n>0, and c, for any natural number k\geq f^*_t(n,c) the following holds. Assume \Lambda is an alphabet with n>0 letters, M is the family of non empty subsets of \{1,\ldots,k\} with \leq t members and V is the set of functions from M to \Lambda and lastly d is a c–colouring of V (i.e., a function with domain V and range with at most c members). Then there is a d–monochromatic V–line, which means that there are w \subseteq \{1,\ldots,k\}, with at least t members and function \rho from \{u\in M: u not a subset of w\} to \Lambda such that letting L=\{\eta\in V:\eta extend \rho and for each s=1,\ldots,t it is constant on \{u\in M:u\subseteq w has s members \}\}, we have d\restriction L is constant (for t=1 those are the Hales Jewett numbers).
• Version 2002-08-01_10 (35p)
Bib entry
@article{Sh:679,
author = {Shelah, Saharon},
title = {{A partition theorem}},
journal = {Sci. Math. Jpn.},
fjournal = {Scientiae Mathematicae Japonicae},
volume = {56},
number = {2},
year = {2002},
pages = {413--438},
issn = {1346-0862},
mrnumber = {1922806},
mrclass = {05D10 (03D20)},
note = {\href{https://arxiv.org/abs/math/0003163}{arXiv: math/0003163}},
arxiv_number = {math/0003163}
}