Galvin nd pinning countable ordinals

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Galvin, Fred and Larson, Jean, Pinning countable ordinals. Fund. Math. 82 (1974/75), 357-361. The image-only scan prints no copyright or license line on its first or last page; the publisher's record (https://www.impan.pl/get/doi/10.4064/fm-82-4-357-361, read 2026-10-02) labels the PDF download "Pobierz zgodnie z CC-BY", which the English site renders "Free download under CC-BY license", a Creative Commons Attribution license with no version or license URL named; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

A map pi: A -> B between well-ordered sets is a pinning map if the image of every subset of A order-isomorphic to A is order-isomorphic to B; alpha -> beta means such a map exists from order type alpha into beta. The abstract (p. 357) states that the paper determines all countable ordinals alpha for which there is such a map into omega^3, that is, alpha -> omega^3, answering a question raised by Specker. Theorem 1 (p. 358) reduces alpha -> beta to the case of indecomposable ordinals, via Cantor normal forms alpha = a_0 + ... + a_m and beta = b_0 + ... + b_n and a one-to-one map f from {0,...,n} into {0,...,m} with a_{f(i)} -> b_i; Theorem 2 (p. 358) gives the main characterization that for alpha < omega_1, omega^alpha -> omega^3 if and only if alpha is decomposable and alpha >= 3, with Theorem 3 and Theorem 8 supplying the two directions and Lemmas 4-7 (Lemma 4 due to Specker) the supporting steps. The paper reworks part of Chapter 2 of Larson's Ph.D. thesis, written under Baumgartner. Its Theorem 9 (p. 360) draws the consequence that for countable alpha, alpha -> (alpha,3)^2 holds only for alpha in {0,1,omega^2} or alpha = omega^{omega^beta} with beta < omega_1, and the introduction (p. 357) offers the conjecture omega^{omega^beta} -> (omega^{omega^beta}, n)^2 for all such beta and n < omega, noting it is proved only for beta = 0 and beta = 1. This bears on Erdos problem 592, on ordinal partition relations of the form alpha -> (alpha, 3)^2 for countable alpha, since by Specker's observation a pinning alpha -> beta carries a negative relation beta -/-> (beta, n)^2 over to alpha.

Source: http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv82i1p24bwm.

Bears on. #592

Results to transcribe.

• Theorem 1 (p. 358): Reduces alpha -> beta to indecomposable ordinals: with Cantor normal forms, alpha -> beta iff a one-to-one index map f exists with a_{f(i)} -> b_i.
• Theorem 2 (p. 358): For alpha < omega_1, omega^alpha -> omega^3 if and only if alpha is decomposable and alpha >= 3.
• Theorem 3 (p. 358): If 3 <= alpha < omega_1 and alpha is decomposable, then omega^alpha -> omega^3.
• Theorem 8 (p. 359): If alpha < omega_1 is indecomposable, then omega^alpha does not pin to omega^3.
• Lemma 6 (p. 359): If alpha < omega_1 is a decomposable limit ordinal then omega^alpha -> omega^3.
• Theorem 9 (p. 360): If alpha < omega_1 and alpha -> (alpha,3)^2, then alpha is in {0,1,omega^2} or alpha = omega^{omega^beta} for some beta < omega_1.