Problem 532

If $\mathbb{N}$ is 2-coloured then is there some infinite set $A\subseteq \mathbb{N}$ such that all finite subset sums

\sum_{n\in S}n

(as $S$ ranges over all non-empty finite subsets of $A$ ) are monochromatic?

Source: erdosproblems.com/532

Number theory
Ramsey theory

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

The site labels the problem PROVED (LEAN) and its curator credits Hindman [Hi74] with the proof, for every finite number of colors. Hindman's theorem (J. Combin. Theory Ser. A 17 (1974), 1--11, refereed) answers the question for every finite coloring. Hindman's paper is in the publisher's open archive: its Theorem 3.1 (p. 9) is the statement for a finite partition of the positive integers, and its proof (Lemmas 2.2--2.12, pp. 2--9, and a compactness argument on p. 9) was read for structure only. The second refereed proof is Baumgartner's note [Ba74] (J. Combin. Theory Ser. A 17 (1974), 384--386), whose Theorem 1 (p. 384) is the statement for $k$ cells of the nonnegative integers and whose Theorem 2, the finite-unions form, is proved in two pages; the theorem statements were checked here and the proof was read for structure only. The site's "(LEAN)" suffix is a catalog label: the theorem is in Mathlib (Hindman.FS_partition_regular, Hindman.exists_FS_of_finite_cover) and an external Lean file derives the site's statement from it; both are cited at pinned commits from their text, nothing was built and no local kernel credit is claimed (Formalization below). Label, sources and field agree. The claim pages Hindman 1974 and Baumgartner 1974 (both accepted on their refereed publications, Hindman's also on the site's credit; the Mathlib proof and the external Lean derivation of the site's statement are formalization links on Hindman's page, not built here) record the results, their postings and their acceptance evidence, and the frontmatter standing derives from them.

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 (received October 1, 1972; published July 1974 per the Crossref record). In the publisher's open archive, 11 pages. Theorem 3.1 with its proof, p. 9; Definition 2.1 and the notation, p. 1; Lemma 2.2, p. 2; Lemma 2.12, p. 9; Corollaries 3.2--3.5 with the closing remark, pp. 10--11; the lemma chain of pp. 3--9 is followed for structure only. Library home: Hindman 1974 and its Hindman 1974, Theorem 3.1 page.
Ba74: Baumgartner, J. E., A short proof of Hindman's theorem. J. Combin. Theory Ser. A 17 (1974), no. 3, 384--386, doi:10.1016/0097-3165(74)90103-4; Theorems 1 and 2, p. 384; Lemmas 1--4, pp. 385--386. Library home: Baumgartner 1974.
Er73: Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), North-Holland (1973), 117--138; p. 122. Library home: Erdos 1973.
Er75b: Erdős, P., Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295--310; p. 305. Library home: Erdős 1975.
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; Section 6, p. 57. Library home: Erdos 1977.
Er80: Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89--115; Section 5, pp. 104--105. Not a source key of this problem on the site; its sentences on the theorem's three proofs are quoted below. Library home: Erdos 1980.

Problem 532

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Proof claims

Discussion