# Sh:679

- Shelah, S. (2002).
*A partition theorem*. Sci. Math. Jpn.,**56**(2), 413–438. arXiv: math/0003163 MR: 1922806 -
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} }