Problem 1199

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claims/: The 2 claim pages of Problem 1199, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that in any $2$-colouring of $\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?

Formulation. The site's wording; the site's page shows no last-edited date. Here $A+A=\{a+b:a,b\in A\}$ includes the doubles $2a$; without them, for the pairwise sums of distinct elements, every finite coloring has such an $A$: Leader and Williams derive this from Ramsey's theorem, and the site's commentary from Hindman's theorem (Problem 532). Owings's original wording ([Ow74], Amer. Math. Monthly problem E 2494, p. 902; the quotation in [HLSXXZ26], p. 1, agrees with it): "Prove or disprove: Given any subset $B$ of $N$, there exists an infinite set $A\subseteq N$ such that $A+A\subseteq B$ or $A+A\subseteq N\setminus B$", where the print's $N$ is the set of natural numbers, with $A+A$ defined on the same page as the set of sums $a_1+a_2$, $a_1,a_2\in A$, so that the doubles are included; the proposal is posed to prove or disprove and states no expected answer. Erdős's versions ask for an infinite sequence with "all the sums $x_i+x_j$ ($i=j$ permitted)" in one class ([Er77c], p. 58; [Er80], p. 104, writes the sums as $a_i+a_j$). Hindman's form ([Hi79], p. 19) asks the same for $r$ cells: "Must there exist some $i<r$ and some sequence $\langle x_n\rangle_{n<\omega}$ of distinct members of $N$ such that $x_m+x_n\in A_i$ whenever $\{m,n\}\subseteq\omega$?" The site's question is $r=2$, and the answer is no for every $r\ge3$ ([Hi79], p. 19: "It is shown in section 2 that the answer is 'no' if $r\ge3$"). The site's source key is [Er80, p. 104].

Status. Open. For three or more colors the answer is no: Hindman [Hi79] (J. Combin. Theory Ser. A 27 (1979), 19--32, refereed) constructed a three-cell partition with no infinite $A$ having $A+A$ monochromatic (Theorem 2.4, p. 21, with its proof on pp. 21--23, followed below); the result rests on the paper, to which the site, [LeWi24], [HLSXXZ26] and Erdős's 1977 and 1980 surveys all attribute it; a different explicit three-coloring recorded in an external Lean file behind the formal-conjectures variant is checked below by an elementary argument. For two colors the paper itself settles only the admissible partitions, those with a class containing, for a fixed difference $d$, arbitrarily long arithmetic progressions $\{x+kd:0\le k\le n\}$ that start at an even integer $x$ (Corollary 2.10, p. 27), recorded as an accepted partial claim on its claim page, and closes its § 2 with the conjecture that the restriction can be removed (p. 28). The three-color Theorem 2.4, which the site credits, answers a variant and settles no instance of the two-color question, so it has no claim page. For two colors in general, a preprint of Huang, Lian, Shao, Xiao, Xu and Zhang ([HLSXXZ26], arXiv:2607.17333, v3 of 29 July 2026) claims an affirmative answer (its Theorem 1.1). No refereed version, independent review or citing paper of that claim was found, and the site had not adopted it (its label was OPEN on 2026-09-18, seven weeks after a thread comment of 1 August 2026 reported it); the claim is recorded as a pending full claim on its claim page, from which the frontmatter standing claimed, proved is derived. For the two-color statement the absence of acceptance evidence is a bounded negative finding of the search, not a certificate of openness.

Source. erdosproblems.com/1199, accessed 2026-09-18: the problem page (OPEN, with the site's note that no finite computation can settle it; no last-edited date; source key [Er80, p.104]; commentary citing [Ow74], [Hi79] and Problem 532; indicators "Formalised statement? Yes" and "OEIS: Possible"), its four-comment discussion thread (three comments of 11 April 2026, one of 1 August 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #1199, https://www.erdosproblems.com/1199, accessed 2026-09-18.

References.

• [Ow74] Owings, J. C., Jr., Problem E 2494, in Elementary Problems: E2492--E2496. Amer. Math. Monthly 81 (1974), no. 8, 901--902, the problem on p. 902 (the site's and the papers' citation; the Crossref record for doi:10.2307/2319455 names the column and its first page, 901). The column prints the proposal only, on p. 902, with no solution; no file of it is held. Library home: owings_1974_e2494_sumset_within_set_or_complement; result page Problem E 2494.
• [Hi79] Hindman, N., Partitions and sums of integers with repetition. J. Combin. Theory Ser. A 27 (1979), no. 1, 19--32, doi:10.1016/0097-3165(79)90004-9 (received April 28, 1977; July 1979 per the Crossref record; zbMATH 0419.05001), free to read in the publisher's open archive; no file of it is held. Cited: the introduction, p. 19; Theorem 2.1 and Definition 2.2, pp. 20--21; Definition 2.3 and Theorem 2.4 with its proof, pp. 21--23; Theorem 2.9 and Corollary 2.10, pp. 26--27; Theorem 2.11 and the closing conjecture, p. 28; Theorem 4.3, p. 31. Library home: hindman_1979_partitions_sums_integers_repetition; result pages Theorem 2.4 and Corollary 2.10.
• [Er80] Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; Section 5, pp. 104--105. 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; Section 6, p. 58. Library home: erdos_1977_problems_results_combinatorial_number_theory_iii.
• [HLSXXZ26] Huang, W., Lian, Z., Shao, S., Xiao, R., Xu, L. and Zhang, S., An affirmative answer to Owings's sumset question. arXiv:2607.17333 (v1 19 July 2026; v3 29 July 2026; 39 pages). An unrefereed preprint; Theorem 1.1, p. 2; Theorem 1.6 and Remark 1.7, p. 4; Section 5.3, p. 35. Library home: huang_2026_affirmative_answer_owings_sumset_question; result page Theorem 1.1.
• [LeWi24] Leader, I. and Williams, K., Monochromatic sumsets in countable colourings of abelian groups. arXiv:2407.03938v1 (4 July 2024); the introduction's Owings sentences, p. 1; Theorem 1, p. 2. Library home: leader_2024_monochromatic_sumsets_countable_colourings_abelian_groups.
• [FSV24] Fernández-Bretón, D., Sarmiento Rosales, E. and Vera, G., Owings-like theorems for infinitely many colours or finite monochromatic sets. Ann. Pure Appl. Logic 175 (2024), 103495; arXiv:2402.13124. Context, cited from its abstract.
• [KLRSSV19] Komjáth, P., Leader, I., Russell, P. A., Shelah, S., Soukup, D. T. and Vidnyánszky, Z., Infinite monochromatic sumsets for colourings of the reals. Proc. Amer. Math. Soc. 147 (2019), 2673--2684; arXiv:1710.07500v3 (12 pages; Problem 1.1, p. 1, and Problem 3.1 in Section 3). Context; not filed.
• [Ko26] Kousek, I., Sharp density conditions for infinite $B+B$ sumsets in abelian groups. arXiv:2607.18132 (v1 20 July 2026; v2 8 August 2026). Context, cited from its abstract.
• [KMRR] Kra, B., Moreira, J., Richter, F. K. and Robertson, D., Problems on infinite sumset configurations in the integers and beyond. Bull. Amer. Math. Soc. 62 (2025), no. 4, 537--574, doi:10.1090/bull/1868; arXiv:2311.06197 (v2 14 May 2025). Context ([HLSXXZ26], p. 3, says its Question 3.8 is Owings's question), cited from its abstract.

Formalization. Statement only here. The file ErdosProblems/1199.lean of formal-conjectures (main on 2026-09-18) declares erdos_1199 : answer(sorry) ↔ ∀ (color : ℕ → Fin 2), ∃ (A : Set ℕ), A.Infinite ∧ ∀ n ∈ (A+A), ∀ m ∈ (A+A), color n = color m under category research open with proof sorry, and the variant erdos_1199.variants.three : ∃ (color : ℕ → Fin 3), ∀ (A : Set ℕ), A.Infinite → ∃ n ∈ (A+A), ∃ m ∈ (A+A), color n ≠ color m under research solved with proof sorry and a formal_proof attribute naming HowieHwong/lean-erdos-proofs Erdos/P1199.lean#L85 at a fixed commit; its docstring attributes the three-color failure to Hindman [Hi79]. That external file (4,950 bytes; import Mathlib; no sorry, no axiom declaration, no #print axioms line) defines logColor n as $\lfloor3\log n/\log4\rfloor\bmod3$ and proves the variant; the argument is checked below. The community database of 9 September 2026 records the problem open (last changed 4 April 2026), the statement formalized since 4 May 2026, no formal proof and an OEIS entry marked "possible". The corpus has built neither Lean file.

Current assessment

The question (the site's formulation). The statement above; OPEN, with the site's note that no finite computation can settle it; no last-edited date. The commentary attributes the conjecture to Owings [Ow74], reports that Hindman [Hi79] showed it false for three colors, and notes that once the doubles $2a$, $a\in A$, are left out, Hindman's theorem gives a positive answer (Problem 532). The thread: on 11 April 2026 a commenter proposed coloring $n$ by the parity of $v_2(n)$, so that $n$ and $2n$ always differ; the site's author replied that $n$ and $2n$ need not both lie in $A+A$, and that $A=\{n\equiv1\pmod4\}$ makes all of $A+A$ blue in that coloring (checked here: $a+b\equiv2\pmod4$ has $v_2(a+b)=1$); the proposer agreed. On 1 August 2026 a comment reported that the six authors of [HLSXXZ26] claim an affirmative answer and linked the arXiv preprint. No reply and no change of label followed by 18 September 2026. The proof-claim tab is empty.

The origin. The Monthly proposal (Problem E 2494 of [Ow74], p. 902) poses the two-class question in the department's "Prove or disprove" form and states no expected answer; the site's attribution of a conjecture to Owings and the 1980 survey's "Ewing conjectured" are their words ([Er77c] speaks of a question of "Ewings"). [Er77c], p. 58, reports that Hindman, answering a question of "Ewings", proved, in a paper to appear in the Journal of Combinatorial Theory, that when the integers are divided into two classes there is always an infinite sequence $x_1<x_2<\ldots$ with "all the sums $x_i+x_j$ ($i=j$ permitted)" in one class; that Hindman had just told Erdős of a possible gap in the proof, which Erdős hoped would be repaired before the survey appeared; and that Hindman had split the integers into three classes $A_1,A_2,A_3$ admitting no such sequence, one class of density $0$, his example satisfying (1) $A_1(x)=\sum_{a_i\in A_1,\,a_i\le x}1<cx^{1/2}$, with the question of improving (1) left open. [Er80], pp. 104--105, writes that "Ewing conjectured" the existence of an infinite sequence $a_1<\cdots$ with all sums $a_i+a_j$, the doubles $2a_i$ included, in one class, calls it "rather annoying that this simple and interesting question is still open", and reports Hindman's preliminary results: the conjecture certainly fails for three classes, and one class can have density $0$, in fact (1) $A_1(x)=\sum_{a_i<x}1<Cx^{1/2}$, with whether (1) is best possible not yet clear. Erdős writes "Ewings" and "Ewing" for Owings. The 1977 passage records only Hindman's warning that the announced two-class proof may have a gap; the paper confirms that the announcement was in error (p. 19: "This author erroneously announced [10] a proof that the answer is 'yes' if $r=2$", [10] being a 1976 Notices abstract). The paper proves the two-cell case only for admissible partitions, those with a cell containing, for a fixed difference $d$, arbitrarily long arithmetic progressions $\{x+kd:0\le k\le n\}$ that start at an even integer $x$ (Definition 2.2, pp. 20--21; Corollary 2.10, p. 27, its claim page), constructs an admissible three-cell partition with no monochromatic $A+A$ (Theorem 2.4, p. 21), and closes its § 2 with "the obvious conjecture" that Theorem 2.9 holds without admissibility (p. 28); [HLSXXZ26], p. 2, adds that "The non-admissible two-color case remained open, as recorded by Hindman and Strauss". The density remark of the surveys is traced to the paper: p. 21 records that an earlier three-cell example "had the property that the density of one cell is 0" and Erdős's question, in a personal communication, "how small one could make the third cell"; the published construction's small cell satisfies $|\{x\in A_0:x<t\}|<n^2$ for $d_n\le t<d_{n+1}$, where $t>2^n$ (p. 23), so it has fewer than $(\log_2t)^2$ elements below $t$; and Theorems 2.4 and 2.9 answer the question (p. 21): for a non-decreasing $f$, every admissible three-cell partition with one $f$-small cell has a cell containing all pairwise sums of some sequence of distinct integers if and only if $f$ is bounded, where $f$-small means covered by intervals $[d_n,d_n+t_n]$ with $t_n\le f_n$ and $d_{n+1}-d_n-t_n\to\infty$ (Definition 2.3). The surveys' bound $cx^{1/2}$ is not printed in the paper.

Three colors: false (Hindman's Theorem 2.4, and an explicit coloring checked here). Theorem 2.4 of [Hi79] (p. 21): "Let $f$ be any unbounded non-decreasing sequence in $N$. There is an admissible partition $\{A_i\}_{i<3}$ of $N$ such that $A_0$ is $f$-small and such that there are no $i<3$ and no sequence $\langle x_n\rangle_{n<\omega}$ of distinct members of $N$ with $x_m+x_n\in A_i$ whenever $\{m,n\}\subseteq\omega$." The pairs $\{m,n\}$ include $m=n$, so the excluded sets are exactly the sums $A+A$, doubles included, of an infinite set $A$, the site's pattern. The partition (pp. 22--23) is by intervals: with increasing sequences $a_n<b_n<c_n<d_n<a_{n+1}$ satisfying $d_n=2b_n$, $c_{n+1}=2d_n$, $a_{n+1}=2c_n$ and $c_n\le2a_n$, the cells are $A_0=\bigcup_{n\ge k}[d_n,a_{n+1})$, $A_1=\bigcup_n[a_n,c_n)$ and $A_2=N\setminus(A_0\cup A_1)$; doubling sends $[d_n,a_{n+1})$, $[a_n,b_n)$, $[b_n,c_n)$ and $[c_n,d_n)$ into $A_2$, $A_2$, $A_0$ and $A_1$ respectively, while the gaps grow so that, for a sequence inside one of the four sets, the sums $x_0+x_n$ for large $n$ avoid the cell holding the doubles. The site, [LeWi24] (p. 1: "it is a simple matter to find a 3-colouring yielding no such monochromatic set"), [HLSXXZ26] (p. 2) and the two Erdős surveys attribute the three-color counterexample to this theorem. The external Lean file behind the collection's variant records a different explicit coloring, which is checked here as an authored argument: color $n\ge1$ by $c(n)=\lfloor\log_4(n^3)\rfloor\bmod3$. Given an infinite $A$, pick $a\in A$ with $a>0$ and then $b\in A$ with $b>4a$; both $a+b$ and $2b$ lie in $A+A$. The ratio $r=2b/(a+b)$ satisfies $8/5<r<2$, so $4<(8/5)^3<r^3<8<16$ and $1<\log_4((2b)^3)-\log_4((a+b)^3)=3\log_4r<2$; two reals whose difference lies strictly between $1$ and $2$ have floors differing by $1$ or $2$, so $c(a+b)\ne c(2b)$. Hence no infinite $A$ has $A+A$ monochromatic under $c$, and the two-color question is the last finite case. The Lean file proves exactly this (logColor, floor_mod_ne, scaled_log_bounds, ratio_cube_bounds, then erdos_1199.variants.three); the corpus has not built it. This coloring is not Hindman's: each of its three classes is the union of the intervals $[4^{k/3},4^{(k+1)/3})$ over $k$ in one residue class modulo $3$ and has positive upper density, while Hindman's $A_0$ has density $0$ (p. 23); both are interval partitions in which doubling moves every cell (an observation made here). The coloring by the parity of $v_2(n)$ that the thread proposed appears on [Hi79], p. 19, as Hindman's example that his 1974 theorem fails "if one allows so much as a single repetition", there defeating a pattern that contains both $x$ and $2x$.

Two colors: the known results. Corollary 2.10 of [Hi79] (p. 27; an accepted partial claim on its claim page): "Let $\{A_i\}_{i<2}$ be an admissible partition of $N$. Then there exist a sequence $\langle x_n\rangle_{n<\omega}$ of distinct members of $N$ and $i<2$ such that $x_m+x_n\in A_i$ whenever $\{m,n\}\subseteq\omega$", the case $A_0=\emptyset$ of Theorem 2.9 (p. 26); the proof of Theorem 2.9 (pp. 26--27) reduces through Lemmas 2.5--2.8 to Ramsey's theorem and to Hindman's theorem in its finite-unions form (Corollary 3.3 of [Hi74]) and has been followed for structure only. The paper also proves the finite version for every finite partition and every number of colors (Theorem 2.1, p. 20, attributed there to Rado and to Deuber) and shows that no analog with a coefficient holds for two cells: for $m\ge2$ some two-cell partition has no sequence of distinct members with all $mx_r+x_s$, $r\ne s$, in one cell (Theorem 2.11, p. 28). The remaining positive partial results are second-hand from [HLSXXZ26], pp. 1--3: Kousek and Radić constructed a three-coloring with syndetic cells and no monochromatic $B+B$ and observed that Owings's question is equivalent to its shifted form (a monochromatic $B+B+t$ for some infinite $B$ and $t\ge0$); and for a two-coloring with no infinite $B$, shift $t$ and color making $B+B+t$ monochromatic, the upper and lower densities of each color class sum to $1$ (their Proposition 6.3, as [HLSXXZ26], p. 3, reports it). [LeWi24] (July 2024) states that the two-coloring case "is unknown" and [KLRSSV19] (2019) that Problem 1.1, Owings's question, "is still unsolved". Variants: for infinitely many colors or finite monochromatic sets, [FSV24]'s abstract reports complete answers for $\kappa,\theta$ both finite and both infinite; [LeWi24]'s Theorem 1 gives, for every abelian group without elements of order $4$, a countable coloring with no monochromatic $\{2x,2y,x+y\}$; over the reals and rational vector spaces the question is Problem 965's territory, as [LeWi24]'s introduction surveys; density analogs (a shifted $B\oplus B$ in every set of positive upper density, and [Ko26]'s density conditions for $B+B$ in abelian groups) are context, not the problem. Erdős's own density strengthening of Hindman's theorem appears on [Er77c], pp. 57--58, and [Er80], p. 105, next to the Owings passages.

The 2026 claim (pending, recorded on its claim page). Theorem 1.1 of [HLSXXZ26] (p. 2): "Let $\mathbb{N}=C_1\sqcup C_2$. Then there exist $i\in\{1,2\}$ and an infinite $B\subseteq\mathbb{N}$ such that $B+B\subseteq C_i$." This is the same statement as the site's question, with a two-cell partition in place of a 2-coloring and $B+B$ in place of $A+A$. The preprint's Theorem 1.6 (p. 4) claims the weighted form, $(m+\ell)B\cup\{mx+\ell y:x,y\in B,x<y\}$ monochromatic for every $m,\ell\in\mathbb{N}$, and its Section 5.3 (p. 35) claims that the three-fold version ($B+B+B$ monochromatic) fails for some two-coloring. The route, named from the abstract's key words and Section 2, is topological dynamics and ultrafilters; the proof of Theorem 1.1 (Section 3, pp. 14--19) argues by contradiction from the hypothesis that no infinite $B$ has $B+B$ monochromatic, builds a coloring with a thick color class and applies Hindman's admissible-partition theorem (Corollary 2.10 of [Hi79]). No step of the argument has been checked for this corpus; the preprint was posted 19 July 2026 (v1) and revised to v3 on 29 July 2026 with the comment "we resolve the weighted form of Owings's sumset question completely". Acceptance evidence: none. No journal record (a Crossref bibliographic query for the title found none), no citing paper (Semantic Scholar's citation list is empty), no site adoption (label OPEN; the thread's report of 1 August 2026 unanswered), no independent review found, and the community database of 9 September 2026 records the problem open. The claim is recorded as a pending full claim on its claim page, from which the frontmatter standing claimed, proved is derived; a refereed publication or a documented independent acceptance would move the claim page to accepted and the standing with it.

Search scope. None of the routes below found a refereed proof or disproof of the two-color statement, a review of the preprint, or a second proof.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures file and the external Lean file at the commits linked above; the community database of 9 September 2026.
• arXiv: the API records of 2607.17333 (v1 and v3, no journal reference) and 2407.03938 (v1 only); the API query all:Owings AND (abs:sumset OR abs:sumsets OR abs:"B+B" OR abs:monochromatic) sorted by date (30 records, titles read; the relevant ones are [HLSXXZ26], [Ko26] and [FSV24]); the API records of 2605.24751, 2311.06197, 2607.18132, 2402.13124 and 1801.09179 (abstracts read).
• Crossref: a bibliographic query for the preprint's title (no record); the records for [Ow74] and [Hi79]. Semantic Scholar: the paper record for arXiv:2607.17333 was not obtained; its citation list (empty). zbMATH Open and OpenAlex records for [Hi79] (no review or abstract text served).
• The primary sources at the pages cited: [HLSXXZ26] pp. 1--2, 4, 14, 18--19 and 35; [LeWi24] pp. 1--2; [Er77c] p. 58 and [Er80] pp. 104--105; [KLRSSV19] p. 1.

Not searched: MathSciNet, Google Scholar, X. Not held: the Kousek--Radić paper and the Hindman--Strauss survey that [HLSXXZ26] cites.

Remaining gaps. (1) The 2026 claim is unreviewed; its acceptance would move its claim page to accepted and the standing with it. (2) The column of [Ow74] prints no solution, and whether the Monthly later printed one is not established. The proof of Hindman's Theorem 2.4 is followed above and the proof of Theorem 2.9 is not checked; the three-coloring checked above is the Lean file's, not Hindman's. (3) The corpus has built neither Lean file. (4) The surveys' display (1), $A_1(x)<cx^{1/2}$, is not printed in [Hi79]: the published construction has the sharper count $A_0(t)<(\log_2t)^2$ (p. 23), and p. 21 mentions an earlier example with a density-zero cell without giving it; which example display (1) describes is not settled.

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
• hindman_1979_partitions_sums_integers_repetition
• hindman_1979_partitions_sums_integers_repetition / corollary_2_10
• hindman_1979_partitions_sums_integers_repetition / theorem_2_1
• hindman_1979_partitions_sums_integers_repetition / theorem_2_11
• hindman_1979_partitions_sums_integers_repetition / theorem_2_4
• huang_2026_affirmative_answer_owings_sumset_question
• huang_2026_affirmative_answer_owings_sumset_question / theorem_1_1
• leader_2024_monochromatic_sumsets_countable_colourings_abelian_groups
• owings_1974_e2494_sumset_within_set_or_complement
• owings_1974_e2494_sumset_within_set_or_complement / problem_e2494