---

Claim. In the reading $\alpha=\omega^\beta$ of Problem 592 (its Formulation), Specker answers the question at every finite exponent $\beta\ge2$. He proves $\omega^2\to(\omega^2,n)^2$ for every finite $n$, so $\beta=2$ has the property, and $\omega^3\not\to(\omega^3,3)^2$. He introduces pinning maps (a map from $\alpha$ into $\beta$ sending every subset of order type $\alpha$ to a set of order type $\beta$), observes that a partition relation $\alpha\to(\alpha,\chi)^2$ passes to every $\beta$ to which $\alpha$ can be pinned, and shows that $\omega^m$ can be pinned to $\omega^3$ for $3\le m<\omega$. Together these give $\omega^m\not\to(\omega^m,3)^2$ for every finite $m\ge3$, so no finite $\beta\ge3$ has the property. The paper is E. Specker, Teilmengen von Mengen mit Relationen, Comment. Math. Helv. 31 (1956/57), 302--314, DOI 10.1007/BF02564361, the site's [Sp57]; the publisher's record gives receipt on 8 October 1956 and the issue date December 1956, and the page name carries the first day of that month, since the record gives no day. The paper is not filed in the library: the statements above follow the account of it in the introduction of Galvin and Larson's Pinning countable ordinals (card) and in Schipperus's history on p. 1196 of Countable partition ordinals (card), not the paper's pages.

Covers. The exponents $\beta=2$ (the property holds) and every finite $\beta\ge3$ (it fails). Not covered: every infinite $\beta$, among them Chang's $\beta=\omega$.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in Commentarii Mathematici Helvetici, volume 31. The site labels the problem OPEN, and its commentary crediting Specker on an open problem is not an acceptance, so no reviewed evidence is listed.