Sh:1258

• Shelah, S. Consistency of square bracket partition relation. Preprint. arXiv: 2601.02923
• Abstract:
  Characteristic earlier results were of the form {\mathrm{CON}}(2^{\aleph_0} \to [\lambda]_{n,2}^2), with 2^{\aleph_0} an ex-large cardinal, in the best case the first weakly Mahlo cardinal.

  Characteristic new results are {\mathrm{CON}} \big( (2^{\aleph_0} = \aleph_m) + \aleph_{\ell} \to [\aleph_k]_{n,2}^2 \big), for suitable k < \ell < m. So we improve in three respects: the continuum may be small (e.g. not a weakly Mahlo), we use no large cardinal, and the cardinals \lambda involved are <2^{\aleph_0} after the forcing.

• Version 2026-01-06 (14p)

@article{Sh:1258,
 author = {Shelah, Saharon},
 title = {{Consistency of square bracket partition relation}},
 note = {\href{https://arxiv.org/abs/2601.02923}{arXiv: 2601.02923}},
 arxiv_number = {2601.02923}
}