Problem 1198

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?

Source: erdosproblems.com/1198

Additive combinatorics
Ramsey theory

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

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).

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.
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.
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 and its Hindman 1974, 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 and its Hindman 1980, 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.

Problem 1198

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