Sh:1249

• Garti, S., Hayut, Y., & Shelah, S. On a problem of Erdös and Hajnal. Preprint. arXiv: 2502.16625
• Abstract:
  We address a question of Erdős and Hajnal about the ordinary partition relation \aleph_{\omega+1}\nrightarrow(\aleph_{\omega+1},(3)_{\aleph_0})^2.

  For \theta=cf(\lambda)<\lambda, assuming 2^\lambda=\lambda^+ they proved the negative relation \lambda^+\nrightarrow(\lambda^+,(3)_\theta)^2 and asked whether the (local instance of) GCH is indispensable. We show that this negative relation is consistent with \lambda being a strong limit and 2^{\lambda}>\lambda^+.

  The result can be pushed down to \aleph_{\omega}.

• Version 2026-04-20 (30p)

@article{Sh:1249,
 author = {Garti, Shimon and Hayut, Yair and Shelah, Saharon},
 title = {{On a problem of Erd{\"o}s and Hajnal}},
 note = {\href{https://arxiv.org/abs/2502.16625}{arXiv: 2502.16625}},
 arxiv_number = {2502.16625}
}