Owings 1974 e2494 sumset within set or complement

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problem_e2494: Owings's 1974 Monthly proposal, the origin of Problem 1199: prove or disprove that every subset B of the natural numbers admits an infinite set A with A + A, doubles included, inside B or inside its complement; posed as a problem, not as a conjecture, and printed without a solution.

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J. C. Owings, Jr., Problem E 2494, in Elementary Problems: E2492--E2496, Amer. Math. Monthly 81 (1974), no. 8 (October), 901--902, the problem on p. 902; JSTOR stable id 2319455, DOI 10.2307/2319455 (the DOI and the Crossref record name the column and its first page, 901). The column is the elementary section of the Monthly's Problems and Solutions department, edited by Emory P. Starke (masthead, p. 901); its proposals E 2492 through E 2497 are by Knuth, Cooper, Owings, Klamkin, Whittekin, and King and Hosford, and p. 902 ends at the head of the solution of E 1085. The proposer is named as "J. C. Owings, Jr., University of Maryland" (p. 902). Cited as [Ow74] on the problem page; the site's citation "Amer. Math. Monthly (1974), 902", which omits the volume, and Hindman's "Amer. Math. Monthly 81 (1974), 902" are confirmed by the page image. The edition read for this card is the version of record as JSTOR serves it; no other version exists. The corpus's other sources cite the proposal second-hand: Hindman's 1979 paper, filed as hindman_1979_partitions_sums_integers_repetition, cites it as its [12] and says its question was "asked for the case $r=2$ by J. C. Owings" (p. 19); the 2026 preprint filed as huang_2026_affirmative_answer_owings_sumset_question quotes the problem (its p. 1), and the quotation agrees with the printed text.

The copy read for this card is JSTOR's scan of the two printed pages: 3 pages, PDF p. 1 a JSTOR cover sheet (title, authors, source, stable URL, access date) and printed pp. 901--902 = PDF pp. 2--3 (printed p. $n$ is PDF p. $n-899$); printed p. 901 carries the end of the preceding article and the department's masthead, and printed p. 902 the six proposals and the head of a solution. The scan has a text layer that reads the prose cleanly and garbles the displays (the sum of E 2492 and the limit of E 2495) and the set symbols of E 2494 (subset and membership signs come out as letters); each page ends in JSTOR's download line, and the file's metadata records only its 2026 assembly. Provenance: the copy was obtained on 2026-09-22 from JSTOR through the library's acquisition, under the library's subscription access, from the stable URL https://www.jstor.org/stable/2319455; 307,134 bytes. The article pages print no copyright line; the JSTOR cover sheet (PDF p. 1) names the publisher as "Taylor & Francis on behalf of the Mathematical Association of America" and states that "Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms", every page prints "All use subject to https://about.jstor.org/terms", and those terms (https://about.jstor.org/terms/, read 2026-10-02) allow use "for research activities, including downloading or printing Content in reasonable amounts for non-commercial, scholarly purposes", prohibit redistribution to non-authorized users, commercial use and systematic distribution, and grant no Creative Commons license, every other right reserved.

Read status: claims checked for Problem E 2494 (printed p. 902, PDF p. 3), read clause by clause on the page image, with the proposer line; the cover sheet (PDF p. 1) and the masthead and submission notes (p. 901, PDF p. 2) were read on the page images for the citation, and the other five proposals on p. 902 on the page image for context only. The column prints proposals only: no solution, editorial comment or reference accompanies E 2494, and whether the Monthly later printed a solution was not established here. Nothing here is independently reviewed.

Contents

• Cover sheet (PDF p. 1). JSTOR's bibliographic sheet: the column title "Elementary Problems: E2492-E2496", eight named authors (the seven proposers and one further name), the source line "Vol. 81, No. 8 (Oct., 1974), pp. 901-902", the publisher line (Taylor & Francis on behalf of the Mathematical Association of America) and the stable URL.
• Printed p. 901 (PDF p. 2, page image). The last paragraphs of an article on history-of-mathematics courses at the University of Toronto, then the department masthead: editor, associate editors, collaborating editors and the University of Maine Problems Group, with the note that proposals go to the editor and that problems from well-known textbooks or generally accessible sources are not appropriate.
• Printed p. 902 (PDF p. 3, page image). The heading "Elementary Problems" with the instruction that solutions go to the Problems Group at the University of Maine before January 31, 1975, then the proposals. E 2494, proposed by Owings: with $N$ the set of natural numbers and, for $A\subseteq N$, $A+A$ the set of all sums $a_1+a_2$ with $a_1,a_2\in A$, the reader is asked to prove or disprove that for every subset $B$ of $N$ there is an infinite set $A\subseteq N$ with $A+A\subseteq B$ or $A+A\subseteq N\setminus B$. The definition of $A+A$ allows $a_1=a_2$, so the doubles $2a$ belong to $A+A$; the proposal does not say whether $0$ is a natural number. It is posed in the Monthly's "Prove or disprove" form and carries no conjectured answer. The other proposals (E 2492, a sum of remainders; E 2493, integers whose divisor sum is a power of $2$; E 2495, a limit; E 2496, a nonsingularity criterion for a square matrix; E 2497, a divisibility in a doubly infinite recurrence) do not bear on the corpus's problems.

Compiled scope

The column is compiled for the one proposal the citing problem consumes, E 2494 (p. 902), read on the page image and paged on problem_e2494. The other proposals are listed for the record and not consumed. The column contains no solution, so the source settles nothing about the answer.

Bears on. #1199: Problem E 2494 (printed p. 902, PDF p. 3) is the origin of the problem, the two-class question the site states. Quoted in the part where the exact wording matters (p. 902): "Prove or disprove: Given any subset $B$ of $N$, there exists an infinite set $A\subseteq N$ such that $A+A\subseteq B$ or $A+A\subseteq N\setminus B$." Taking $B$ as one color class of a two-coloring, this is the site's question, with $A+A$ defined on the same page to include the doubles. The proposal is a prove-or-disprove problem with no stated expectation; the site's "A conjecture of Owings" and the 1980 Erdős survey's "Ewing conjectured" (p. 104; the 1977 survey, p. 58, writes "Answering a question of Ewings") are their words, not the Monthly's. The column prints the proposal only and leaves the problem's status where the page has it.

Results.

• Problem E 2494 (p. 902): for every $B\subseteq N$, is there an infinite $A\subseteq N$ with $A+A\subseteq B$ or $A+A\subseteq N\setminus B$? Posed to prove or disprove; no solution printed here.

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