Problem 1198

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claims/: The 1 claim page of Problem 1198, one per claimant's result; the problem's standing derives from them.

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Statement. If $\mathbb{N}$ is $2$-coloured then must there exist an infinite set $A=\{a_1<\cdots\}$ such that all expressions of the shape

$$\prod_{i\in S_1}a_i+\cdots+\prod_{i\in S_k}a_i,$$

for disjoint $S_1,\ldots,S_k$ (excluding the trivial expressions $a_i$) are the same colour?

Formulation. The site's wording as accessed 2026-09-18 (page last edited 17 April 2026). The expressions are Erdős's "multilinear expressions formed from the $a$'s" ([Er80], p. 104): sums of $k\ge1$ products over pairwise disjoint nonempty finite sets $S_1,\ldots,S_k$ of indices, excluding the single-term products $a_i$ themselves (the case $k=1$, $|S_1|=1$; the thread's reading is $\max(|S_1|,k)\ge2$). The elements $a_i$ need not lie in the class: Erdős separately notes that "A further complication arises if we also insist that $a_1,a_2,\ldots$ should also belong to the same class" ([Er80], p. 104), and the thread's first comment records that the site's wording was revised on this point in April 2026. The case in which every $S_i$ is a singleton is Hindman's theorem (Theorem 3.1 of [Hi74], p. 9; Problem 532), as the commentary says; the finite question for sums and products of distinct elements is Problem 172.

Status. Disproved. The status-defining source is a refereed theorem of Smith [Sm95] (J. Combin. Theory Ser. A 72 (1995), no. 1, 77--94), not held here, from which two comments of the site's thread (16 April 2026) deduce a $2$-coloring of $\mathbb{N}$ under which no infinite $A$ has all its multilinear expressions in one class; the deduction is rewritten below as an authored derivation from Smith's statements as the thread quotes them, and it is elementary given those statements. The site adopted the disproof, crediting the counterexample to Smith [Sm95] in its commentary (page last edited 17 April 2026), and the community database records it (24 April 2026). The claim page Smith 1995 records the theorem, the thread's deduction and the acceptance evidence; the frontmatter standing is derived from it. Smith's paper is not held, although the Crossref record carries the publisher's open-archive license, and the statements used from Smith are the thread's restatements, second-hand and marked as such below. The thread's first comment judged Hindman's 1980 theorem [Hi80] not to settle the problem. Erdős's own guess was "no", "but no counterexample is in sight" ([Er80], p. 104).

Source. erdosproblems.com/1198, accessed 2026-09-18: the problem page (DISPROVED, with the site's note that it is solved in the negative; last edited 17 April 2026; source key [Er80, p.104]; commentary citing [Hi74], [Sm95] and Problems 532 and 172, with additional thanks to Aron Bhalla and Wouter van Doorn), its four-comment discussion thread (all of 16 April 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #1198, https://www.erdosproblems.com/1198, accessed 2026-09-18.

References.

• [Er80] Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; Section 5, p. 104. Library home: erdos_1980_survey_problems_combinatorial_number_theory.
• [Er77c] Erdős, P., Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Math. 626, Springer (1977), 43--72; p. 58, the 1977 form of the question. Library home: erdos_1977_problems_results_combinatorial_number_theory_iii.
• [Hi74] Hindman, N., Finite sums from sequences within cells of a partition of $N$. J. Combin. Theory Ser. A 17 (1974), no. 1, 1--11, doi:10.1016/0097-3165(74)90023-5. The singleton case: Theorem 3.1, p. 9, gives for every finite partition of the positive integers a cell and a sequence all of whose finite sums of distinct terms lie in that cell; the paper treats sums only, never products. Open access in the publisher's archive. Library home: hindman_1974_finite_sums_sequences_within_cells_partition_n and its theorem_3_1 page; Baumgartner's short proof is compiled on Problem 532's page.
• [Sm95] Smith, G. L., Partitions and ($m$ and $n$) sums of products. J. Combin. Theory Ser. A 72 (1995), no. 1, 77--94, doi:10.1016/0097-3165(95)90029-2 (published October 1995 per the Crossref record; zbMATH 0868.05005, whose review text is not served). Not held; the Crossref record lists the publisher's open-archive license (from 2013), so a browser download should be possible. Its Theorem 3.17 and its two-cell theorem for $(n,m)=(1,2)$ are restated second-hand from the thread.
• [Sm01] Smith, G. L., Partitions and ($m$ and $n$) sums of products---two cell partition. SIAM J. Discrete Math. (2001), doi:10.1137/S0895480198349014. Context (a later two-cell paper by the same author, found among the works citing [Sm95]); not held, not read.
• [Hi80] Hindman, N., Partitions and sums and products---two counterexamples. J. Combin. Theory Ser. A 29 (1980), no. 1, 113--120, doi:10.1016/0097-3165(80)90052-7. Theorem 2.14, p. 117, is the two-coloring result the thread's first comment restates: a two-cell partition $\{J_0,J_1\}$ of $\mathbb{N}$ such that no infinite $A\subseteq J_t$ has all its finite products and pairwise sums in $J_t$. Open access in the publisher's archive. Library home: hindman_1980_partitions_sums_products_two_counterexamples and its theorem_2_14 page.
• [Gr26] Griffin, C., Infinite sum-product configurations in parallel. arXiv:2605.24751v1 (23 May 2026). A lead citing [Sm95]; abstract read only.

Formalization. None. No file ErdosProblems/1198.lean existed in formal-conjectures (main) on 2026-09-18; the site's page says "Formalised statement? No", and the community database (9 September 2026) records the problem disproved (last updated 24 April 2026), not formalized, with no formal proof.

Current assessment

The question (site formulation of 2026-09-18). The statement above; DISPROVED, with the site's note that it is solved in the negative, last edited 17 April 2026. The commentary records that the case in which every $S_i$ is a singleton is the Graham--Rothschild conjecture proved by Hindman [Hi74] (Problem 532), repeats Erdős's guess that the answer is no although no counterexample was in sight, credits the counterexample to Smith [Sm95] and points to Problem 172. The thread, four comments of 16 April 2026, oldest first: (1) the account Woett, later marked by its author as superseded by the updated problem statement and remarks: the page had misquoted Hindman's 1980 paper, whose relevant result is a $2$-coloring of $\mathbb{N}$ such that for every infinite $A$ the set $A\cup\{\prod_{a\in S}a:S\subseteq A\text{ finite}\}\cup\{a+b:a\ne b\in A\}$ is not monochromatic (the seven-color result reducing the middle set to pairwise products), which does not settle the problem because it requires $A$ itself in the class; the comment reads Erdős's "multilinear expressions" as excluding the $a_i$ and expects a counterexample to exist. (2) Woett again, crediting ChatGPT with pointing to Smith's paper: with $n=1$ and $m=2$ in Smith's notation there is a $2$-coloring $X\sqcup Y$ of $\mathbb{N}$ such that for any infinite $B=\{b_1<b_2<\cdots\}$ and $C=\{c_1<c_2<\cdots\}$ some finite product $b_{i_1}\cdots b_{i_r}\notin X$ and some sum of two finite products $c_{j_1}\cdots c_{j_s}+c_{k_1}\cdots c_{k_t}\notin Y$ (all indices distinct); the comment then takes an arbitrary infinite $A=\{a_1<a_2<\cdots\}$ and puts $b_i=a_{2i}a_{2i+1}$ and $c_i=a_i$, which it says yields a counterexample to the problem; marked as addressed by a site update. (3) The account AronBhalla, restating Smith: with $SP_n((x_t))$ the set of sums of $n$ increasing block-products of a sequence, Theorem 3.17 gives, for any distinct $m_1,\ldots,m_r$, a partition of $\mathbb{N}$ into $r$ cells that separates the corresponding $SP_{m_i}$-systems; taking $r=2$, $m_1=2$ and $m_2=3$ yields a $2$-coloring of $\mathbb{N}$ in which no color class contains both an $SP_2((x_t))$ and an $SP_3((y_t))$, and the comment observes that this answers the multilinear formulation directly, since $SP_2((a_t))$ and $SP_3((a_t))$ consist of multilinear expressions. (4) Woett agreeing. The proof-claim tab is empty. The community's AI-contributions wiki (last updated 30 June 2026) lists the 16 April 2026 contribution as partial results found; the deduction on the site is the commenters' own.

The origin. [Er80], p. 104: "Some time ago, I thought of the following fascinating problem: Divide the integers into two classes. Is it true that there always is an infinite sequence $a_1<\cdots$ so that all the multilinear expressions formed from the $a$'s are all in the same class. One would perhaps guess that the answer must be 'no' but no counterexample is in sight." The next paragraph poses a "much weaker conjecture", an infinite sequence with all sums $a_i+a_j$ and products $a_ia_j$ in one class, adds that "A further complication arises if we also insist that $a_1,a_2,\ldots$ should also belong to the same class", records Graham's $252$ and Hindman's $990$ for the pattern $x,y,x+y,xy$ and ends "(Recently Hindman found some very interesting counterexamples.)" [Er77c], p. 58, has the 1977 form: after asking for a sequence with all finite sums $\sum\varepsilon_ia_i$ and all finite products $\prod a_i^{\varepsilon_i}$ in one class ("At this moment the problem is open"), "More generally one can ask: Is there an infinite sequence $a_1<a_2<\ldots$ so that all the multilinear expressions formed from the a's are in the same class? One would perhaps guess that the answer is no but no counter example is in sight."

Smith's theorem (second-hand). Smith's paper was not read; the zbMATH record serves no review text and the Crossref and OpenAlex records carry no abstract, so the statements below are the thread's restatements and nothing more. (a) Theorem 3.17, as comment (3) 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 $n$ increasing block-products of a sequence $x_1<x_2<\cdots$; the reading used below is that an "increasing block-product" sum is $\prod_{t\in F_1}x_t+\cdots+\prod_{t\in F_n}x_t$ with finite nonempty index sets $F_1<\cdots<F_n$, every element of $F_i$ below every element of $F_{i+1}$. (b) The two-cell statement for $(n,m)=(1,2)$, as comment (2) states it: a $2$-coloring $X\sqcup Y$ of $\mathbb{N}$ such that for any infinite $B$ and $C$ some finite product of distinct elements of $B$ is outside $X$ and some sum of two finite products of distinct elements of $C$ (all indices distinct) is outside $Y$.

Authored derivation (made here; valid given the statements above). Suppose $A=\{a_1<a_2<\cdots\}$ is infinite and all its multilinear expressions lie in one color class $\mathcal C$. From (a) with $r=2$, $m_1=2$, $m_2=3$: every element of $SP_2((a_t))$ is $\prod_{F_1}a_t+\prod_{F_2}a_t$ with disjoint blocks $F_1<F_2$, a multilinear expression with $k=2$, and every element of $SP_3((a_t))$ is one with $k=3$; so $\mathcal C$ contains both an $SP_2$-set and an $SP_3$-set, which the coloring of (a) forbids. From (b): put $b_i=a_{2i}a_{2i+1}$ and $c_i=a_i$; $B=\{b_i\}$ is infinite and increasing. A finite product $b_{i_1}\cdots b_{i_r}$ is $\prod_{t\in S_1}a_t$ over a set $S_1$ of $2r\ge2$ indices, a multilinear expression with $k=1$ and $|S_1|\ge2$, not a trivial $a_i$; a sum $c_{j_1}\cdots c_{j_s}+c_{k_1}\cdots c_{k_t}$ with all indices distinct is a multilinear expression with $k=2$. If $\mathcal C=X$, every product of the $b$'s lies in $X$, against (b); if $\mathcal C=Y$, every such sum of two products lies in $Y$, against (b). Either way no such $A$ exists, so the answer to the site's question is no. Both routes use only that the configurations named above are multilinear expressions of $A$ other than single terms; nothing about Smith's proof is used or checked. By contrast, when every $S_i$ is a singleton the expressions are the finite sums, which Hindman's theorem makes monochromatic for some infinite $A$; the products are what destroy the partition regularity.

Hindman's theorem (first-hand). Theorem 2.14 of [Hi80] (p. 117): "It is not the case that there exist $t$ in $\{0,1\}$ and $A$ in $[J_t]^\omega$ such that $FP(A)\cup PS(A)\subseteq J_t$", where $\{J_0,J_1\}$ is an explicit partition of $\mathbb{N}$ defined on p. 115 from the positions of binary digits, $[X]^\omega$ is the set of infinite subsets of $X$, $FP(A)=\{\prod F:F\in\mathrm{fin}(A)\}$ over the finite non-empty $F\subseteq A$, and $PS(A)=\{x+y:\{x,y\}\in[A]^2\}$ (Definition 2.1, p. 114). This is the result the thread's first comment restates in its own notation, and the printed statement is equivalent to that restatement. The comment judged it not to settle the problem because the theorem requires $A$ itself inside the cell. Under the site's wording before April 2026, with the $a_i$ required in the class, the theorem applies directly.

Search scope (2026-09-18 UTC). None of the routes below found a text of Smith's theorem beyond the thread's quotations, a dispute of the deduction, or a second source for the disproof.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory listing (no file); the community database as of 9 September 2026.
• Crossref record for [Sm95] (JCTA 72 (1995), no. 1, 77--94, October 1995; open-archive license from 17 July 2013); the zbMATH Open record (0868.05005; "contents unavailable due to conflicting licenses") and the OpenAlex record (no abstract; open access at the publisher); the publisher's full-text download link, which did not serve the file.
• Semantic Scholar's list of works citing [Sm95] (six records: [Sm01], two editions of a textbook on Ramsey theory on the integers, a 2009 topology paper, a 2002 survey chapter on the algebra of $\beta S$, and [Gr26]; titles read, [Gr26]'s abstract read through the arXiv API); the arXiv API query abs:"sums of products" AND (abs:partition OR abs:colouring OR abs:coloring) AND abs:Hindman (two records, neither on this problem).
• The primary sources, to the depth stated above: [Er80] p. 104 and [Er77c] p. 58; [Hi74] p. 9; [Hi80] pp. 113--115 and 117--118.

Not searched: MathSciNet, Google Scholar, X. Not held: [Sm95], [Sm01].

Remaining gaps. (1) Smith's theorem is second-hand: the definition of $SP_n$ and the two statements are as the thread quotes them, and the derivation from them is valid only for those statements; the reopening condition is Smith's text read (JCTA 72 (1995), 77--94; the open-archive license suggests a browser download). (2) No formal statement exists; the derivation from Smith's statements is authored here. (3) Theorem 3.1 of Hindman's 1974 paper (p. 9) and Theorem 2.14 of his 1980 paper (p. 117) are read first-hand, the latter's proof (pp. 117--118) for structure only and not checked. (4) [Sm01], a two-cell paper by the same author, was not read and may give a more direct two-cell theorem.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1977_problems_results_combinatorial_number_theory_iii
• erdos_1980_survey_problems_combinatorial_number_theory
• baumgartner_1974_short_proof_hindman_theorem
• baumgartner_1974_short_proof_hindman_theorem / theorem_1
• erdos_1976_problems_results_combinatorial_number_theory_ii
• erdos_1976_problems_results_combinatorial_number_theory_ii / problem_p290
• graham_rothschild_1971_ramseys_theorem_n_parameter_sets
• graham_rothschild_1971_ramseys_theorem_n_parameter_sets / question_9_ii
• hindman_1974_finite_sums_sequences_within_cells_partition_n
• hindman_1974_finite_sums_sequences_within_cells_partition_n / theorem_3_1
• hindman_1980_partitions_sums_products_two_counterexamples
• hindman_1980_partitions_sums_products_two_counterexamples / theorem_2_14