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Claim. Let $\{A_0,A_1\}$ be an admissible partition of the positive integers $N$. Then there are $i<2$ and a sequence $\langle x_n\rangle_{n<\omega}$ of distinct members of $N$ with $x_m+x_n\in A_i$ whenever $\{m,n\}\subseteq\omega$. This is Corollary 2.10 of N. Hindman, Partitions and sums of integers with repetition (p. 27), the case $A_0=\emptyset$ of his Theorem 2.9 (p. 26). The pairs $\{m,n\}$ include $m=n$, so the infinite set $A=\{x_n:n<\omega\}$ has $A+A$, doubles included, inside one cell: the statement of Problem 1199 for admissible two-colorings.

Covers. The admissible two-colorings: those in which one class contains, for some $d\in N$ and every $n$, a progression $\{x+kd:k\le n\}$ with $x$ even (Definition 2.2, pp. 20--21). The non-admissible two-colorings are the open remainder, which the pending 2026 claim addresses. The same paper's three-cell counterexample (Theorem 2.4, p. 21) answers a variant with three colors and settles no instance of the two-color question, so it has no claim page; the problem page records it.

Acceptance. Refereed: N. Hindman, J. Combin. Theory Ser. A 27 (1979), no. 1, 19--32, received 28 April 1977, published in the July 1979 issue per the Crossref record. The record gives no day, so the day in the page name is a placeholder. The evidence is not reviewed: the site labels the problem OPEN, and its commentary credits the paper only with the three-color result.

Read depth. The statements of Corollary 2.10, Theorem 2.9 and Definition 2.2 were checked; the proof of Theorem 2.9 (pp. 26--27), which runs through Lemmas 2.5--2.8 to Ramsey's theorem and the finite-unions form of Hindman's theorem, was followed for structure only. Nothing here is independent review.

Depends on. Hindman 1974 (accepted), for the finite-unions form of Hindman's theorem (Corollary 3.3 of the 1974 paper) that Lemma 2.5 applies.