Sh:679

• Shelah, S. (2002). A partition theorem. Sci. Math. Jpn., 56(2), 413–438. arXiv: math/0003163 MR: 1922806
• Abstract:
  We deal with some relatives of the Hales-Jewett Theorem with primitive recursive bounds.

  We prove primitive recursive bounds for the polynomial variant of van der Waerden’s Theorem. That is, for a suitable sequence \bar p = \langle p_\ell(x) : \ell < \ell_*\rangle of polynomials and c \in \mathbb{N}, we have

  If \langle A_\ell : \ell < c\rangle is a partition of [1,n] then for some a < n and i < {r} and natural number {x /_r}, we have \{a + p_\ell(x) : \ell < \ell_*\} \subseteq A,

  where n is computed from \bar p and c by a primitive recursive function.

  Minor edits were made to this work in June-July 2026.

• Version 2002-08-01_10 (35p)

@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}
}