Baumgartner (1989): Remarks on partition ordinals

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baumgartner_1989_remarks_partition_ordinals: Records the chapter's identifiers, its URL, and the statements its zbMATH review reports; the chapter itself is not held.

main_theorem: States that Martin's axiom for aleph_1 dense sets makes omega_1 omega and omega_1 omega^2 partition ordinals, alpha -> (alpha, n)^2 for every finite n, as the zbMATH review reports it, with the elementary step to the relation of Problem 1171.

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James E. Baumgartner, Remarks on partition ordinals. In: Set theory and its applications (Toronto, ON, 1987), Lecture Notes in Mathematics 1401, Springer, Berlin, 1989, pp. 5--17. DOI 10.1007/BFb0097328; Zbl 0703.03027; MR 1031762. Cited as [Ba89b] on the problem page.

The folder holds no folder-name PDF: the chapter is paywalled, no open copy was found (the publisher's chapter page served neither an abstract nor a preview on 2026-09-27), and the repository makes no purchases, so the folder-name Markdown file is the source record itself and records the URL and what the zbMATH review states (the library's no-PDF shape). The statements below are the review's account of the chapter, not a reading of it.

A partition ordinal is an ordinal $\alpha$ with $\alpha\to(\alpha,n)^2$ for every finite $n$. According to the review, §1 surveys what is known about partition ordinals, §2 treats Ramsey near-orderings, and §3 proves that Martin's axiom for $\aleph_1$ dense sets, $\mathrm{MA}_{\aleph_1}$, implies that $\omega_1\omega$ and $\omega_1\omega^2$ are partition ordinals. The review contrasts this with the Erdős--Hajnal theorem that the continuum hypothesis gives $\alpha\not\to(\alpha,3)^2$ for both ordinals, so the partition property of $\omega_1\omega$ is independent of ZFC.

A refereed paper states the §3 theorem in the same form: the introduction of Chen, Garti and Weinert, after their Fact 1.1, says that Baumgartner proved in this chapter that $\mathrm{ZFC}+\mathrm{MA}_{\aleph_1}$ implies $\omega_1\omega\to(\omega_1\omega,n)^2$ for all natural numbers $n$. The catalog page for Problem 1171 and the deposit of Gao cite the chapter for the case $n=3$ only.

Relation to Problem 1171. The chapter does not state the catalog relation $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$. It follows from the §3 theorem by an elementary step recorded on the result page: with $n$ the finite Ramsey number for triangles in $k$ colors, a $(k+1)$-coloring of $[\omega_1\omega]^2$ has a color-$0$ homogeneous set of type $\omega_1\omega$ or an $n$-element set colored from the other $k$ colors, hence a monochromatic triangle, and $\omega_1\omega$ is an initial segment of $\omega_1^2$. Since $\mathrm{MA}_{\aleph_1}$ is consistent relative to ZFC, the catalog relation holds in a model of ZFC for every finite $k$. That bridging step is author-recorded here and not independently reviewed; the color reduction lemma of Gao's deposit is the only written form of it found.

Read status. Unread: the chapter's text was not obtained. The consumed statement, the §3 theorem, was checked against the zbMATH review (Zbl 0703.03027, read through the zbMATH Open API) and against the paper of Chen, Garti and Weinert, which state it in the same form; the chapter's own theorem numbering is unknown, so the result page carries a descriptive name.

Bears on. #1171.

Results.

• Main theorem (§3): $\mathrm{MA}_{\aleph_1}$ implies that $\omega_1\omega$ and $\omega_1\omega^2$ are partition ordinals.