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Claim. There is a $2$-coloring of $\mathbb{N}$ under which no infinite set $A=\{a_1<a_2<\cdots\}$ has all its multilinear expressions, the sums $\prod_{i\in S_1}a_i+\cdots+\prod_{i\in S_k}a_i$ over pairwise disjoint nonempty finite index sets other than the single terms $a_i$, in one color class; so the answer to Problem 1198 is no. The coloring comes from Smith's Theorem 3.17 (Partitions and ($m$ and $n$) sums of products, J. Combin. Theory Ser. A 72 (1995), no. 1, 77--94), as the site's thread states it: for distinct $m_1,\ldots,m_r$ there is a partition of $\mathbb{N}$ into $r$ cells such that no cell contains both an $SP_{m_i}$-set and an $SP_{m_j}$-set for $i\ne j$, where $SP_n((x_t))$ is the set of sums of $n$ products over increasing blocks of a sequence $x_1<x_2<\cdots$. With $r=2$, $m_1=2$ and $m_2=3$, every element of $SP_2((a_t))$ and of $SP_3((a_t))$ is a multilinear expression of $A$ with two or three terms, so a class containing all multilinear expressions of $A$ would contain both sets, which the partition forbids. The thread's other route, from Smith's two-cell statement for one product against a sum of two products through the substitution $b_i=a_{2i}a_{2i+1}$, gives the same answer. Both deductions use only that the configurations named above are multilinear expressions other than single terms; they are written out as an authored derivation on the problem page.

Depends on. Nothing in this wiki; the result is Smith's theorem together with an elementary deduction, both second-hand from the site's thread and rewritten on the problem page.

Dating. The page is dated by the issue month of the journal record (J. Combin. Theory Ser. A 72 (1995), no. 1, October 1995, per the Crossref record; the day in the page name is a placeholder. The deduction to this problem was posted in two comments of the site's thread on 16 April 2026.

Acceptance. Reviewed: the site's curator, T. F. Bloom, labels the problem DISPROVED and credits the counterexample to Smith in the problem's commentary, revised on 17 April 2026 after two commenters of the thread deduced it on 16 April 2026 (page and thread as of 2026-09-18); the curator neither submitted nor co-wrote the result and is independent of the author, and the community database records the problem disproved (24 April 2026). Refereed: J. Combin. Theory Ser. A 72 (1995), no. 1, 77--94, which covers Smith's theorem, not the deduction to this problem. The deduction is elementary given the statements as the thread gives them and nothing in it is independently reviewed in this corpus.

Read depth. The paper is not held, although the Crossref record carries the publisher's open-archive license; the zbMATH, Crossref and OpenAlex records serve no review text or abstract. Nothing of it was read; the definition of $SP_n$, Theorem 3.17 and the two-cell statement are exactly as the thread quotes them, and the deduction is valid for those statements. Reopening condition: Smith's text read at Theorem 3.17 and at the two-cell statement, which may also show whether the author's later two-cell paper (SIAM J. Discrete Math., 2001) gives a more direct theorem.