Partitioning Vector Spaces

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main_theorem: Constructs a subset meeting every infinite arithmetic progression while containing no three-element arithmetic progression.

remark_p233: Records Baumgartner's stronger final assertion for each fixed positive integer k, for which the paper omits the modified proof.

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Source. James E. Baumgartner, Partitioning vector spaces, Journal of Combinatorial Theory, Series A 18 (1975), 231–233; received 1974-05-07. DOI. The copy read for this card, the publisher's PDF of printed pp. 231–233, was inspected page by page on 2026-09-06 and read again against the card and the result pages on 2026-10-08. It prints "Copyright © 1975 by Academic Press, Inc. All rights of reproduction in any form reserved." in the footer of printed p. 231, every other right reserved.

The unnumbered Theorem on p. 231 constructs a subset of any vector space over $\mathbb Q$ that meets every one-sided infinite arithmetic progression and contains no three distinct points in arithmetic progression. Taking the vector space to be $\mathbb R$ over $\mathbb Q$ directly disproves Problem 199. The introductory paragraph attributes a conditional predecessor to unpublished work of R. O. Davies under the continuum hypothesis; Baumgartner's proof does not assume that hypothesis. The Davies attribution is recorded as Baumgartner's report, not as a separately inspected source.

Proof scope. The main theorem page reconstructs the full argument on pp. 231–233: a countable collection of coefficient patterns, their transfer to arbitrary finite subsets of an ordered basis, and all three cases excluding a three-term progression. The use of an ordered vector-space basis and a countably infinite basis subset is an explicit external choice dependency. The "still stronger theorem" on p. 233 is recorded separately with its exact quantifiers. The paper omits its modified proof, and this compilation does not supply one.

Relation to Problem 198. No Sidon-set theorem is explicitly stated in this paper. The successive selection of points beyond all previously used coefficient magnitudes is the relevant method behind the public attribution of the integer construction as implicit in Baumgartner's work. The integer construction adds its own Sidon argument; the main theorem's exclusion of three-term progressions alone is not the Sidon property. The historical source record preserves the contradictory positive assertion in the 1979 Erdős–Graham survey and the limits of the attribution.

Bears on.

• Problem 199: the Theorem with $V=\mathbb R$ over $\mathbb Q$ gives a set of reals with no three-term arithmetic progression whose complement contains no infinite arithmetic progression, a negative answer to the question as posed.
• Problem 198: the paper states no result about Sidon sets. The problem's page on erdosproblems.com regards the integer construction recorded there, a Sidon set meeting every infinite arithmetic progression, as implicit in this paper; the paper itself does not state it.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.