Problem 1211

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claims/: The 1 claim page of Problem 1211, one per claimant's result; the problem's standing derives from them.

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Statement. Let $\mathbb{N}=A\cup B$ where $A$ and $B$ are disjoint. Let $S(A)$ be those integers which are the sum of finitely many distinct elements of $A$, and similarly for $S(B)$.

If

$$\overline{\delta}(X)=\limsup_{x\to \infty}\frac{1}{\log x}\sum_{\substack{n\in X\\ n<x}}\frac{1}{n}$$

is the upper logarithmic density of $X$ then how large must

$$\max(\overline{\delta}(S(A)),\overline{\delta}(S(B)))$$

be?

Formulation. The site's wording (page last edited 8 April 2026). $S(A)$ is the set of subset sums of $A$, written $\Sigma(A)$ by Conlon, Fox and Pham and $A^+$ or $A^{(\infty)}$ by Erdős; the sum over $n<x$ in the site's density and over $a\le x$ in the paper's give the same $\limsup$. The question asks for the least value of the maximum over all partitions, the constant $c$ of the site's commentary and $c_2$ of the paper, and the paper's Theorem 1 gives both the value and a partition attaining it. The site's commentary attributes to [Er80] both the bounds $1/2\le c<1$ and the expectation $c>1/2$; the passage on printed p. 113 of that survey says that the maximum "can be less than $1$, but I expect it to be greater than $\frac12$" and states no lower bound; the bound $c\ge1/2$ is claimed as proved on p. 54 of the Mysore paper ("I proved that ... for at least one $i$, $A_i^{(\infty)}$ has upper density $1$ and upper logarithmic density $\ge\frac12$. The proof again uses our Lemma, the details will not be given"); the Congressus Numerantium paper only expects the maximum to exceed $\frac12$ (its (25), p. 39), and its Theorem 3 (p. 38) is the interval statement from which, by Conlon, Fox and Pham's p. 3, the bound easily follows, with a proof Erdős postponed and never published. The example the site gives for $c<1$, the class of the $n$ with $\lfloor\log_4\log n\rfloor$ even and its complement, is Erdős's block construction of the Mysore paper ($n_{i+1}=n_i^4$, $A_1=\bigcup_i[n_{2i},n_{2i+1})$) and of display (24) of the Congressus paper; the paper writes it with $\lfloor\log_4\log_2n\rfloor$ and calls it "essentially the special case where $r=2$ and $b=4$" of its $\lfloor\log_b\log n\rfloor$ coloring, and the value $14/15$ is its computation for that block structure.

Status. Solved, in the site's label for a determination question. The status-defining source is Theorem 1 of Conlon, Fox and Pham: for every partition of $\mathbb N$ into two classes the larger upper logarithmic density of the subset sums is at least $(2+\sqrt3)/4\approx0.93301$, and the coloring by the parity of $\lfloor\log_{2+\sqrt3}\log n\rfloor$ attains it, so the minimum is $c_2=(2+\sqrt3)/4$. The paper appeared in Mathematika 68 (2022), no. 4, 1292--1301 (published online 10 October 2022, per the Crossref record), a refereed journal; the text cited here is the arXiv version v3 of 22 September 2022, and the journal text was not compared. The site accepted the result (SOLVED, last edited 8 April 2026), its curator crediting the paper with the value in the commentary. The claim page Conlon, Fox and Pham 2021 records the theorem, its postings and the acceptance evidence; the frontmatter standing is derived from it. Erdős's expectation $c>1/2$ holds, and his claim that the value cannot exceed $3/4$ was wrong, his example giving $14/15$.

Source. erdosproblems.com/1211, accessed 2026-09-18 at 05:24 UTC: the problem page (SOLVED, with the site's note that the resolution is neither a proof nor a disproof; last edited 8 April 2026; source keys [Er80, p. 113], [Er82c], [Er82d]; commentary on [Er80] and [CFP22]; OEIS "Possible"), its empty discussion thread and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #1211, https://www.erdosproblems.com/1211, accessed 2026-09-18.

References.

• [CFP22] Conlon, D., Fox, J. and Pham, H. T., The upper logarithmic density of monochromatic subset sums. Mathematika 68 (2022), no. 4, 1292--1301; DOI 10.1112/mtk.12167; arXiv:2105.15195v3 (22 September 2022, 9 pp.). Theorem 1 on p. 2, the definitions on pp. 1--2, Lemma 2 and Theorem 3 on p. 3, Remark 5 on p. 4, Conjecture 10 on p. 9 (preprint pages). Library home: conlon_2022_upper_logarithmic_density_monochromatic_subset_sums; result page theorem_1.
• [Er80] Erdős, P., A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115; printed p. 113. Library home: erdos_1980_survey_problems_combinatorial_number_theory.
• [Er82c] Erdős, P., Miscellaneous problems in number theory. Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, 1981), Congr. Numer. 34 (1982), 25--45; Part II, Theorem 3 and display (23) on printed p. 38 and displays (24)--(26) on p. 39. Library home: erdos_1982_miscellaneous_problems_number_theory.
• [Er82d] Erdős, P., Some new problems and results in number theory. In Number theory (Mysore, 1981), Lecture Notes in Math. 938, Springer (1982), 50--74; the site's reference text. The Rényi archive's scan (users.renyi.hu/~p_erdos/1982-32.pdf, 25 pp.) carries the passage at printed p. 54 (PDF p. 5). Library home: erdos_1982_some_new_problems_results_number_theory (the card carries the row for this problem). It is the paper's reference [3].
• [CFP21] Conlon, D., Fox, J. and Pham, H. T., Subset sums, completeness and colorings. arXiv:2104.14766v1 (2021); its Theorem 6.1 is the input Theorem 3 of [CFP22]. Library home: conlon_2021_subset_sums_completeness_colorings.
• [SzVu06] Szemerédi, E. and Vu, V. H., Long arithmetic progressions in sumsets: thresholds and bounds. J. Amer. Math. Soc. 19 (2006), 119--169; its Theorem 7.1 is the published alternative input named in Remark 5 of [CFP22]. Not held.

Formalization. None found. No file ErdosProblems/1211.lean exists in google-deepmind/formal-conjectures (main; the directory was listed in full). The community database (teorth/erdosproblems) records the problem solved (last updated 4 April 2026), not formalized, formal status unformalized, OEIS "possible" and no formal-proof URL. The site's "Formalised statement?" indicator reads "No".

Current assessment

The question (site formulation, accessed 2026-09-18). The statement above; SOLVED, last edited 8 April 2026. The commentary records Erdős's remark in [Er80] that one of $S(A)$ and $S(B)$ has upper density $1$ while both may have lower density $0$; writes $c$ for the least possible value of the maximum and attributes to [Er80] the bounds $1/2\le c<1$ together with the expectation $c>1/2$; gives the example of the $n$ with $\lfloor\log_4\log n\rfloor$ even, for which, by the computation in [CFP22], both subset-sum sets have upper logarithmic density $14/15$; and credits Conlon, Fox and Pham [CFP22] with $c=(2+\sqrt3)/4\approx0.93301$, the upper bound witnessed by the $n$ with $\lfloor\log_b\log n\rfloor$ even for $b=2+\sqrt3$. The thread and the proof-claim tab are empty.

The origins. [Er80], printed p. 113, calls the question "only a few days old": Erdős divides the integers into two classes $n_1<\cdots$ and $m_1<m_2<\cdots$, writes $N_1<N_2<\cdots$ and $M_1<M_2<\cdots$ for the integers that are distinct sums of the $n$'s and of the $m$'s, and says it is easy to see that one of the two sum sequences has upper density $1$ while both may have lower density $0$; then: "It is not clear to me at present how large $\limsup\frac1{\log x}\max\bigl(\sum_{N_i<x}\frac1{N_i},\sum_{M_i<x}\frac1{M_i}\bigr)$ must be. It is easy to see that it can be less than $1$, but I expect it to be greater than $\frac12$", followed by the definition of the upper logarithmic density. [Er82c], Part II, p. 38, with $A^+$, $B^+$ the sets of distinct sums, repeats that one of $A^+$, $B^+$ has upper density $1$ and states the stronger "Theorem 3. There is an absolute constant $c$ and an infinite sequence $n_1<n_2<\cdots$ so that for every $i$ every $n_i<m<cn_i^2$ belongs entirely to $A^+$, (respectively to $B^+$)", adding that apart from the value of $c$ this is easily seen to be best possible. Then display (23) (p. 38), $\max(\bar d_\ell(A^+),\bar d_\ell(B^+))<1$, and on p. 39 the classes $A=\{m:2^{4^{2k}}<m\le2^{4^{2k+1}},\ k=0,1,\ldots\}$ (display (24)) and its complement; "I am sure that (25) $\max(\bar d_\ell(A^+),\bar d_\ell(B^+))>\frac12$", then "It will probably not be difficult to prove (25), but at the moment I do not see how to determine (26) $\min_{A,B}\max(\bar d_\ell(A^+),\bar d_\ell(B^+))=c$ and I postpone the proof of Theorem 3 until I can settle (26). The proof of Theorem 3 and probably (25) is routine, perhaps the proof of (26) is not so trivial." [Er82d], p. 54 (the archive copy): "I proved that if $\bigcup_{i=1}^kA_i$ is the set of all integers then for at least one $i$, $A_i^{(\infty)}$ has upper density $1$ and upper logarithmic density $\ge\frac12$"; the proof, he says, again uses his Lemma and its details are not given; he is "not quite sure if $\frac12$ is best possible here" but finds it easy to see that the value cannot exceed $\frac34$, by the classes $A_1=\bigcup_i[n_{2i},n_{2i+1})$ with $n_{i+1}=n_i^4$ and its complement $A_2$. Display (26) of [Er82c] is the site's question; Theorem 3 of [Er82c] and the claim of [Er82d] were never proved in print (the paper's p. 3: "a weaker version of this lemma, from which the bound $c_r\ge1/2$ easily follows, was previously claimed by Erdős [4, Theorem 3], though the proof of this statement was never published"; its footnote 1 on p. 1: "Erdős incorrectly implies in [3] that in his construction the upper logarithmic density of both $\Sigma(A_i)$ is at most $3/4$").

Status-defining source. Theorem 1 of [CFP22] (p. 2): with $c_r=\min_{\mathbb N=A_1\sqcup\cdots\sqcup A_r}\max_i\bar d_\ell(\Sigma(A_i))$, "For any integer $r\ge2$, $c_r$ is at most $(1-\frac1{2b_0})(1+\frac1{2rb_0-r})$, where $b_0$ is the unique root of the polynomial $b^r-2rb+r-1$ with $b>1$, and this is tight for $r=2$, where $c_2=(2+\sqrt3)/4\approx0.93301$." The site's quantity is $c_2$. The upper bound (pp. 1--2) generalizes Erdős's coloring: give $n$ the value of $\lfloor\log_b\log n\rfloor$ modulo $r$; since the nonzero subset sums of $[m,n]$ lie in $[m,\binom{n+1}2]$, each class's subset sums have upper logarithmic density at most $\delta_r(b)=(1-\frac1{2b})(1-b^{-r})^{-1}$, minimized at the root $b_0$ of $b^r-2rb+r-1$. For $r=2$ the root above $1$ of $b^2-4b+1$ is $b_0=2+\sqrt3$ and $\delta_2(b_0)=(\sqrt3/2)/(1-b_0^{-2})=(2+\sqrt3)/4$. The lower bound $c_2\ge(2+\sqrt3)/4$ is the paper's main contribution: Lemma 2 (p. 3) gives, for every partition of $\mathbb N\cap[N,eN)$ into $r$ classes, a class whose subset sums contain every integer of $[CN,C'N^2]$, proved from Theorem 3, which is Theorem 6.1 of the authors' preprint [CFP21] (Remark 5, p. 4, says Theorem 7.1 of Szemerédi and Vu's published paper can replace it), and Section 3 turns the lemma into the bound through an auxiliary coloring and the Brouwer fixed-point theorem (pp. 4--8). Acceptance evidence: the refereed Mathematika publication (Crossref record: volume 68, issue 4, pages 1292--1301, published online 10 October 2022) and the site's curator's credit of the value to the paper under the label SOLVED. Coverage: the compiled text covers Theorem 1, the definitions and the upper-bound argument (pp. 1--2) and the statements of Lemma 2, Theorem 3, Remark 5 and Conjecture 10; the lower-bound proof (Section 3, pp. 4--8) is not independently reviewed, and nothing is independently reviewed in this corpus. The text cited is the arXiv preprint, so every locator above is a preprint page.

Erdős's example. The classes of [Er82d] are blocks $[n_{2i},n_{2i+1})$ with $n_{i+1}=n_i^4$; in the scale $L=\log x$ the $j$th block occupies $[\ell_j,4\ell_j)$ with $\ell_{j+1}=4\ell_j$. The subset sums of a block of consecutive integers fill the integers from the block's least element to about the square of its largest, so in the logarithmic scale the block $[\ell,4\ell)$ contributes the interval $[\ell,8\ell)$ to its class's subset sums, while everything the class has below the block reaches only to about $2\cdot4^{-1}\ell=\ell/2$ (the square of the previous block's top, $\ell/4$). The class therefore misses $(\ell/2,\ell)$ before each of its blocks. Taking $L=8\ell$ at the end of a block's contribution, the missed length below $L$ is $\ell/2+\ell/32+\cdots=(\ell/2)(16/15)=8\ell/15$, which is $1/15$ of $L$, and the covered proportion is $14/15$; the proportion falls between the ends of consecutive contributions, so the upper logarithmic density of each class's subset sums is $14/15$, as the paper and the site state, and not $3/4$. This elementary block count confirms the figure; it carries no independent review.

What remains open around the problem. Conjecture 10 of [CFP22] (p. 9): for every $r\ge3$ the upper bound of Theorem 1 is the value of $c_r$; the authors "were unable to establish the optimality of our upper bound for $c_r$ without additional assumptions" (their method applies when the auxiliary coloring is cyclic, which every two-coloring is). This concerns partitions into three or more classes and is not the site's question.

Search scope. None of the routes below found a dispute of Theorem 1, a second determination of $c_2$, or progress on Conjecture 10.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory listing at the pinned commit (no file); the community database of 2026-09-18.
• arXiv API: the record of 2105.15195 (three versions, v3 of 22 September 2022, no journal reference carried) and the search abs:"logarithmic density" AND abs:"subset sums" (no records).
• Crossref: a bibliographic query for the paper's title (the Mathematika record above; the DOI 10.1112/mtk.12158 named elsewhere as this paper's is a different Mathematika article and is not used here).
• OpenAlex: the record of [CFP22] (no citing works) and the two works citing [CFP21], of which [CFP22] is one; Semantic Scholar's search endpoint answered HTTP 429 and was not retried.
• The Rényi archive: its index and its scan of the 1982 Mysore paper of the site's key [Er82d] (p. 54).
• The primary sources: [CFP22] pp. 1--4 and 9; [Er80] p. 113; [Er82c] pp. 38--39; [Er82d] p. 54.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not compared: the Mathematika text. Not held: [SzVu06].

Remaining gaps. (1) The source cited is the preprint; the refereed text was not compared, so the locators are preprint pages and any change in the published version is unrecorded. (2) The lower-bound proof (Section 3, pp. 4--8) is not independently reviewed; the compiled coverage is the statement, the upper-bound argument and the elementary block count above. (3) Erdős's Theorem 3 of [Er82c] and the $\ge1/2$ claim of [Er82d] have no published proof; the paper's Lemma 2 supersedes them. (4) The site's commentary attributes the lower bound $1/2\le c$ to [Er80], whose passage states only the expectation $c>1/2$; recorded above, not resolved with the site.

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_1982_miscellaneous_problems_number_theory
• erdos_1982_miscellaneous_problems_number_theory / display_26
• erdos_1982_miscellaneous_problems_number_theory / theorem_3
• erdos_1980_survey_problems_combinatorial_number_theory
• erdos_1982_some_new_problems_results_number_theory
• erdos_1982_some_new_problems_results_number_theory / claim_p54
• erdos_1982_some_new_problems_results_number_theory / lemma_p53
• conlon_2022_upper_logarithmic_density_monochromatic_subset_sums
• conlon_2022_upper_logarithmic_density_monochromatic_subset_sums / theorem_1