Problem 362

../

claims/: The 2 claim pages of Problem 362, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $A\subseteq \mathbb{N}$ be a finite set of size $N$. Is it true that, for any fixed $t$, there are

$$\ll \frac{2^N}{N^{3/2}}$$

many $S\subseteq A$ such that $\sum_{n\in S}n=t$?

If we further ask that $\lvert S\rvert=l$ (for any fixed $l$) then is the number of solutions

$$\ll \frac{2^N}{N^2},$$

with the implied constant independent of $l$ and $t$?

Formulation. The site's wording (page last edited 27 December 2025). Write $f_A(t)=\#\{S\subseteq A:\sum_{n\in S}n=t\}$. The first question asks for an absolute constant $c$ with $f_A(t)\le c\,2^N/N^{3/2}$ for every $N$, every $N$-element $A\subseteq\mathbb N$ and every $t$; the second asks for an absolute $c'$ with $\#\{S\subseteq A:|S|=l,\ \sum S=t\}\le c'2^N/N^2$ for all $N$, $A$, $l$ and $t$. The two questions have separate sources and separate standing here. The sources work with distinct real numbers: Erdős's 1965 survey with $k$ distinct reals and $F(k)$ the maximal multiplicity, Sárközy and Szemerédi with arbitrary distinct positive reals and the maximum over all real $t\ge0$, Stanley with sets of distinct reals, Halász with $n$ vectors $\mathbf a_k=(a_k,1)$ in the plane, $|a_k-a_{k'}|\ge1$ for $k\ne k'$, the $2^n$ signed sums $\sum_k\varepsilon_k\mathbf a_k$ ($\varepsilon_k=\pm1$) and the largest number of them in an open unit ball. Positive integers are positive reals, so the Sárközy--Szemerédi bound applies to the page's sets $A$ (an authored one-line reading), Stanley's exact maximizers are stated for the class they range over, and the passage from Halász's signed sums to the page's fixed-size counts is the authored reduction under The second question below. If $\mathbb N$ is read to include $0$ and $0\in A$, then $S$ and $S\cup\{0\}$ have the same sum and sizes differing by one, so each count for $A$ is the sum of two counts of the same kind for $A\setminus\{0\}$, a set of $N-1$ positive integers, and both bounds hold for $A$ with a larger absolute constant (an authored line; Stanley's Corollary 5.1 with $\zeta=1$ gives the first count's exact maximum in that case). The site's label PROVED (LEAN) refers to the Lean development described under Formalization.

Status. PROVED (LEAN), on the site's label, which the two refereed sources support, one for each question. The derived frontmatter standing is solved, proved: the page lists the two questions as its parts, and each is settled by an accepted partial claim page, [[problems/number_theory/E0362/claims/1965_01_01_sarkozy_szemeredi|Sárközy and Szemerédi 1965]] settling the first question and Halász 1977 the second, so the two claims together settle every part, and both questions are answered yes below. A Lean development that declares itself a formalization of both results is linked from both claim pages and described under Formalization; it was not built here and supplies no formalized evidence. First question: yes, by the Satz of Sárközy and Szemerédi (Acta Arith. 11 (1965), 205--208, refereed): for every $\varepsilon>0$ and $n>n_0(\varepsilon)$, any $n$ distinct positive reals have $\max_tf(t)<(1+\varepsilon)(8/\sqrt\pi)2^n/n^{3/2}$; for the finitely many $N\le n_0(1)$ the trivial $f_A(t)\le2^N$ gives the bound with the constant $n_0(1)^{3/2}$ (an authored line), so $f_A(t)\ll2^N/N^{3/2}$ with an absolute constant. The order is sharp: $A=\{1,\ldots,N\}$ has $\max_tf(t)>c_32^N/N^{3/2}$ (the paper's remark), and Stanley's Corollary 5.1 (SIAM J. Algebraic Discrete Methods 1 (1980), 168--184, refereed) identifies the exact maximum over $N$ distinct positive reals as the middle coefficient of $(1+q)(1+q^2)\cdots(1+q^N)$, attained by $\{1,\ldots,N\}$, and his Corollary 5.3 the maximum over all sets of $N$ distinct reals, attained by $\{-\lfloor(N-1)/2\rfloor,\ldots,\lfloor N/2\rfloor\}$, the site's set. Second question: yes, by Halász's Theorem 2 and the remark that follows it (Period. Math. Hungar. 8 (1977), 197--211, refereed): for $n$ vectors $\mathbf a_k\in\mathbb R^d$ with $|\mathbf a_k-\mathbf a_{k'}|\ge1$ for $k\ne k'$ such that, for some $\delta>0$ and every unit vector $\mathbf e$, at least $\delta n$ of them satisfy $|(\mathbf a_k,\mathbf e)|\ge1$, at most $c(\delta,d)2^nn^{-1-d/2}$ of the $2^n$ signed sums $\sum_k\varepsilon_k\mathbf a_k$ lie in any open unit ball, and the remark records "a conjecture of Erdős (oral communication), confirmed by Theorem 2: $N\le c2^nn^{-2}$ if $\mathbf a_k=(a_k,1)$, $|a_k-a_{k'}|\ge1$, i.e., if in the above result of Sárközi and Szemerédi the number of $+$ signs in $\mathbf S$ is also fixed", the multidimensional theorem the site's commentary credits and its consequence. For the page, the subsets $S\subseteq A$ with $|S|=l$ and $\sum S=t$ are the sign vectors whose sum $\sum_k\varepsilon_k(a_k,1)$ is the point $(2t-\sum A,,2l-N)$, and translating $A$ so that its middle element is $0$, which leaves the fixed-size counts unchanged, supplies the condition on $\delta$ with $\delta=1/4$ for $N\ge6$, so the count is at most $c(1/4,2)2^N/N^2$ with an absolute constant (an authored reduction, recorded under The second question below; $N\le5$ is covered by the trivial bound $2^N$). The conjecture itself is Erdős's (1965, with the constant independent of the subset size, his $t$, the page's $l$; 1973, display (8.8), with the constant independent of $t$, $l$, $n$ and the sequence). Halász's printed proof of Theorem 2 is a one-paragraph modification (p. 208) of his proof of the probabilistic Theorem 4, not checked here. The site's label agrees with the two refereed sources.

Source. erdosproblems.com/362, accessed 2026-09-18: the problem page (PROVED (LEAN), with the site's note that the answer is affirmative and the proof has been checked in Lean; last edited 27 December 2025; source keys [Er65], [Er73, p. 129], [ErGr80, p. 59]; commentary citing [SaSz65], [St80] and [Ha77]; the formalization indicator then showing no formalized statement, superseded by the files described under Formalization), its one-comment discussion thread (2 November 2025) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #362, https://www.erdosproblems.com/362, accessed 2026-09-18.

References.

• [SaSz65] Sárközi, A. and Szemerédi, E., Über ein Problem von Erdös und Moser. Acta Arith. 11 (1965), no. 2, 205--208, DOI 10.4064/aa-11-2-205-208 (received 26 November 1964; the paper prints the first author as Sárközy). The Satz, p. 205. Library home: sarkozi_1965_uber_ein_problem_von_erdos_und; result page satz.
• [St80] Stanley, Richard P., Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168--184, DOI 10.1137/0601021 (received 1 June 1979). The abstract, p. 168; Corollaries 5.1 and 5.3, pp. 178--179. Library home: stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner; result pages corollary_5_1 and corollary_5_3.
• [Ha77] Halász, G., Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hungar. 8 (1977), no. 3--4, 197--211, DOI 10.1007/BF02018403 (received 29 January 1976, p. 211). Theorem 1, p. 197; Theorem 2 and the remark on Erdős's conjecture, p. 198; Theorem 4, p. 199; the proofs of Theorems 1--3, p. 208. Library home: halasz_1977_estimates_concentration_function_combinatorial_number_theory_probability; result page theorem_2.
• [Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181--189. Printed pp. 183--184: the definitions (11), the conjectures (12) and (13), the fixed-cardinality conjecture and Theorem 1. Library home: erdos_1965_extremal_problems_number_theory.
• [Er73] Erdős, P., Problems and results on combinatorial number theory. A Survey of Combinatorial Theory (Fort Collins 1971), North-Holland (1973), Chapter 12, 117--138. Printed p. 129: displays (8.7) and (8.8). Library home: erdos_1973_problems_results_combinatorial_number_theory.
• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980). Printed p. 59. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Ng12] Nguyen, H. H., A new approach to an old problem of Erdős and Moser. J. Combin. Theory Ser. A 119 (2012), DOI 10.1016/j.jcta.2012.01.003; arXiv:1112.0755. Not held; cited for its abstract, as context on the stability of Stanley's optimal sets.
• Not held: Katona's paper cited "im Druck" by [SaSz65] and as [Kat (66)] by [ErGr80]; van Lint, Proc. Amer. Math. Soc. 18 (1967), 182--184 (Stanley's [42], which misprints the volume as 19, and the monograph's [Lint (67)]); Peck, Studies in Applied Math., "to appear" (Stanley's [35]).

Formalization. The (LEAN) suffix of the site's label refers to a Lean development. The file ErdosProblems/362.lean of formal-conjectures (at the linked commit of 19 September 2026, the only one to touch the file by 2026-10-07) declares, under category research solved and with sorry bodies, erdos_362 : answer(True) ↔ ∃ C : ℝ, ∀ (A : Finset ℕ) (t : ℕ), A.Nonempty → ({S ∈ A.powerset | ∑ n ∈ S, n = t}.card : ℝ) ≤ C * 2 ^ A.card / (A.card : ℝ) ^ (3 / 2 : ℝ) for the first question and erdos_362.variants.fixed_card, the same statement with A.powersetCard l in place of A.powerset and the divisor (A.card : ℝ) ^ 2, for the second, each with a formal_proof attribute naming line 2541 of src/latest/ErdosProblems/Erdos362.lean in Boris Alexeev's repository plby/lean-proofs, at the commit the two claim pages' links pin. That file (2,559 lines, headed leanprover/lean4:v4.33.0 mathlib v4.33.0, imports Mathlib and the repository's ErdosProblems.Erdos487) declares itself a formalization of a solution to Problem 362, names András Sárközy, Endre Szemerédi and Gábor Halász as informal authors and Codex and GPT-5.6 Sol as formal authors, cites [SaSz65] for the first estimate and [Ha77] for the fixed-cardinality estimate, and closes with theorem erdos_362 at that line, the conjunction of the two bounds with absolute constants over all nonempty finite $A\subseteq\mathbb N$ (the first with $\sqrt N^{\,3}$ as its divisor), followed by #print axioms. The file contains no sorry, no axiom declaration and no native_decide. This description rests on the text of the files at the pinned commits: nothing was built or audited here, and no kernel credit is claimed. The file is linked from both claim pages, its first conjunct as a formalization of Sárközy and Szemerédi's result and its second of Halász's. On 2026-09-18 formal-conjectures had no file for the problem (its directory FormalConjectures/ErdosProblems/ had 673 entries), the discussion and proof-claim pages linked no Lean file, and the community database (teorth/erdosproblems) at its revision of that day listed formal_status Lean, as of its last update on 24 August 2026, with the statement not formalized and no formal-proof URL; its revision of 2026-10-06 lists the statement as formalized, as of its last update on 19 September 2026.

Current assessment

The question (site formulation). The two questions above; PROVED (LEAN); last edited 27 December 2025; source keys [Er65], [Er73, p. 129], [ErGr80, p. 59]. The commentary records that the bound of Erdős and Moser [Er65] for the first question carried an extra factor $(\log n)^{3/2}$, that Sárközy and Szemerédi [SaSz65] removed it and so answered the first question yes, that Stanley [St80] showed the count to be largest for $A=\{-\lfloor(N-1)/2\rfloor,\ldots,\lfloor N/2\rfloor\}$, and that Halász [Ha77] answered the second question yes as a consequence of a more general result in higher dimensions. The one comment (2 November 2025) supplies the locators [Er73, p. 129] and [ErGr80, p. 59] and objects that the site had stated Stanley's result in the monograph's form, which the commenter finds wrong or at best vague, because a count over subsets of varying sizes is not preserved under translation, and points to the corollaries of Section 5 of Stanley's paper (the comment calls it Chapter 5) as the correct statements; the site marks the comment as addressed, and the commentary, names Stanley's set. There are no proof claims. The community database record, at its revision of 2026-09-18, lists the status proved (Lean) as of its last update on 24 August 2026, and the informal status proved as of its last update on 31 August 2025.

Erdős's statements. [Er65], printed p. 183: "In the second part of this paper I now give some results together with their proofs which we obtained jointly with L. Moser. Let $a_1<a_2<\cdots<a_k$ be $k$ distinct real numbers. Denote by $f(n;a_1,a_2,\cdots,a_k)$ the number of solutions of $n=\sum_{i=1}^k\epsilon_ia_i$, $\epsilon_i=0$ or $1$, (11) and put $F(k)=\max_{n,a_1,\cdots,a_k}f(n;a_1,\cdots,a_k)$." Then, on p. 184: "It seems likely that $f(n;a_1,\cdots,a_k)$ assumes its maximum if $n=0$ and the $a$'s are $0,\pm1,\pm2,\cdots$. In other words (12) $F(k)=f(0;-[\frac k2],-[\frac{k-2}2],\cdots,0,1,\cdots,[\frac{k-1}2])$, but we have not been able to prove (12). It may be possible to obtain an explicit formula for the right side of (12) but we have not succeeded in doing so. It is easy to see that $F(k)>c_12^k/k^{3/2}$ and in fact it is not hard to show that the right side of (12) is $>c_12^k/k^{3/2}$. We conjecture that (13) $F(k)<c_22^k/k^{3/2}$. A still sharper conjecture than (13) would be that the number of solutions of $n=\sum_{i=1}^k\epsilon_ia_i$, $\sum_{i=1}^k\epsilon_i=t$, $\epsilon_i=0$ or $1$ is less than $c_32^k/k^2$ ($c_3$ is independent of $t$). We were unable to prove (13), but prove the weaker THEOREM 1. $F(k)<c_42^k\bigl(\frac{\log k}k\bigr)^{3/2}$." The proof (pp. 184--186) rests on a Lemma bounding the multiplicity of subset sums of a sequence in which no term is a sum of others, via Sperner's theorem. So (13) is the first question, the "still sharper conjecture" is the second, and (12), the extremal set, is what Stanley proved. [Er73], printed p. 129: "Let $a_1<\cdots<a_n$ be $n$ distinct numbers; L. Moser and I proved that the number solutions [sic] of (see [II]) $t=\sum_{i=1}^n\varepsilon_ia_i$, $\varepsilon_i=0$ or $1$, (8.7) is less than $c2^n(\log n)^{3/2}/n^{3/2}$. We conjectured that it is in fact less than $c2^n/n^{3/2}$ (which apart from the value of $c$ is best possible). Sárközi and Szemerédi [1965] proved this conjecture. It seems that the number of solutions of $t=\sum_{i=1}^n\varepsilon_ia_i$, $\sum_{i=1}^n\varepsilon_i=l$, (8.8) is less than $c2^n/n^2$ (where $c$ is an absolute constant independent of $t$, $l$, $n$ and our sequence). (8.8) has never been proved. It is likely that for $n=2m+1$ the number of solutions of (8.7) is largest when the $a$'s are the integers in $(-m,+m)$, but this has never been proved (Van Lint [1967])." ("the number solutions" as printed.) [ErGr80], printed p. 59: "For the sequence $0<a_1<\ldots<a_n$, let $F_n(t)$ denote the number of solutions of $\sum_{i=1}^n\varepsilon_ia_i=t$, $\varepsilon_i=0$ or $1$. Erdös and Moser (see [Kat (66)]) proved that $F_n(t)<\frac{c2^n}{n^{3/2}}(\log n)^{3/2}$; they conjectured that the factor $(\log n)^{3/2}$ could be omitted and this was proved by Sárközy and Szemerédi [Sár-Sz (65)]. Stanley [Stan (xx)] recently showed that $\max F_n(t)$ is assumed if the $a_i$'s form an arithmetic progression and $t=\frac12\sum_{i=1}^na_i$ (see also [Lint (67)])." The monograph does not state the second question, and its Stanley sentence is the paraphrase the thread faults.

The first question (Sárközy and Szemerédi). [[../library/number_theory/sarkozi_1965_uber_ein_problem_von_erdos_und/satz|The Satz]] (p. 205), with $0<a_1<\cdots<a_n$ arbitrary reals and $f(t)$ the number of solutions of $\sum_{i=1}^n\varepsilon_ia_i=t$, $\varepsilon_i\in\{0,1\}$: "Es sei $\varepsilon>0$ eine beliebige Zahl. Dann ist für $n>n_0(\varepsilon)$ $\max_{0\le t<+\infty}f(t)<(1+\varepsilon)\frac8{\sqrt\pi}\cdot\frac{2^n}{n^{3/2}}$." The introduction recalls the Erdős--Moser bound $c_1\frac{2^n}{n^{3/2}}\log^{3/2}n$, the conjecture $c_22^n/n^{3/2}$ and the lower bound $\max_{t\le n^2}f(t)>c_3(2^n/n^{3/2})$ for $a_i=i$. Read depth: claims checked; the indirect proof (pp. 205--208: a Lemma modifying a theorem of Katona, proved from Sperner's theorem, applied to the solution sets split between the $[n/2]$ smallest and the remaining elements) was read for structure and not checked. Acceptance: refereed publication in Acta Arithmetica (Crossref record); Erdős's own 1973 and 1980 reports; the site. For the page: an $N$-element $A\subseteq\mathbb N$ is a set of distinct positive reals, so $f_A(t)<(1+\varepsilon)(8/\sqrt\pi)2^N/N^{3/2}$ for $N>n_0(\varepsilon)$, and $f_A(t)\le2^N$ covers the finitely many smaller $N$ (authored line).

The exact maximizers (Stanley). [[../library/number_theory/stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner/corollary_5_1|Corollary 5.1]] (p. 178): for a set $A$ of distinct reals with $\nu$ negative elements, $\zeta\in\{0,1\}$ zeros and $\pi$ positive elements, and subsets $B_1,\ldots,B_r$ of $A$ whose element sums take at most $k$ distinct values, $r$ does not exceed the sum of the $k$ middle coefficients of $G_{\nu\zeta\pi}(q)=2^\zeta(1+q)(1+q^2)\cdots(1+q^\nu)\cdot(1+q)(1+q^2)\cdots(1+q^\pi)$, with equality for $\{-1,\ldots,-\nu\}\cup\{1,\ldots,\pi\}$, with $0$ added when $\zeta=1$. [[../library/number_theory/stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner/corollary_5_3|Corollary 5.3]] (p. 179): for $n$ distinct reals, with $\nu=[(n-1)/2]$ and $\pi=[n/2]$, $r$ does not exceed the sum of the $k$ middle coefficients of $2(1+q)(1+q^2)\cdots(1+q^\nu)\cdot(1+q)(1+q^2)\cdots(1+q^\pi)$, with equality for $A=\{-\nu,-\nu+1,\ldots,\pi\}$; "The actual conjecture [13, (12)] of Erdös and Moser is equivalent to the case $k=1$, and $n$ odd, of Corollary 5.3." Consequences for the page (authored, one line each): for $A\subseteq\mathbb N$ with $|A|=N$, Corollary 5.1 with $\nu=\zeta=0$ and $k=1$ gives $f_A(t)\le$ the middle coefficient of $\prod_{i=1}^N(1+q^i)$, the number of subsets of $\{1,\ldots,N\}$ with the central sum, attained by $A=\{1,\ldots,N\}$, a quantity of order $2^N/N^{3/2}$ by the Sárközy--Szemerédi bound and the $c_32^n/n^{3/2}$ remark; Corollary 5.3 with $k=1$ gives the maximum over all $N$-sets of distinct reals, attained by ${-\lfloor(N-1)/2\rfloor,\ldots,\lfloor N/2\rfloor}$, the site's sentence, a set outside $\mathbb N$ since it contains $0$ and negative numbers; the count is not invariant under translating $A$, which is the thread's point against the monograph's "arithmetic progression" paraphrase. Read depth: claims checked; the proofs (Theorem 3.1, property S of the Bruhat-order posets $W^J$ from the hard Lefschetz theorem; Proposition 2.5, on products of varieties with cellular decompositions; and Lemma 5.2) were not checked. Acceptance: refereed publication (Crossref record); Nguyen [Ng12] calls it "the ultimate result" and proves stability of the optimal sets (abstract only; a lead by identifier).

The second question (Halász). [[../library/number_theory/halasz_1977_estimates_concentration_function_combinatorial_number_theory_probability/theorem_2|Theorem 2]] (p. 198), with $\mathbf a_k$ ($k=1,\ldots,n$) vectors of $\mathbb R^d$, the $2^n$ sums $\mathbf S=\sum_{k=1}^n\varepsilon_k\mathbf a_k$ ($\varepsilon_k=+1$ or $-1$) and $N=\max_{\mathbf y\in\mathbb R^d}\sum_{|\mathbf S-\mathbf y|<1}1$ (p. 197), and with the condition of Theorem 1 (p. 197), "there exists a constant $\delta>0$ such that for any $|\mathbf e|=1$ one can select at least $\delta n$ vectors $\mathbf a_k$ with $|(\mathbf a_k,\mathbf e)|\ge1$": "If in addition to the condition of Theorem 1 also $|\mathbf a_k-\mathbf a_{k'}|\ge1$ ($k\ne k'$), then $N\le c(\delta,d)2^nn^{-1-d/2}$", the constant depending only on $\delta$ and $d$. The remark after it (p. 198): "The order of magnitude can be attained for quite different configurations: for the lattice points in a ball around the origin of radius $\sim c(d)n^{1/d}$ or choosing any $(d-1)$-dimensional extremal configuration and translating it orthogonally by a fixed large vector. This latter example is suggested by a conjecture of Erdős (oral communication), confirmed by Theorem 2: $N\le c2^nn^{-2}$ if $\mathbf a_k=(a_k,1)$, $|a_k-a_{k'}|\ge1$, i.e., if in the above result of Sárközi and Szemerédi the number of $+$ signs in $\mathbf S$ is also fixed. This question was the starting point of our investigations in higher dimensions." The "above result" is the paper's placing of the Sárközy--Szemerédi Satz (p. 198) as the $d=1$ case of $N\le c(d)2^nn^{-3/2}$ under $|\mathbf a_k-\mathbf a_{k'}|\ge1$, "somewhat weaker ... in that they take $\mathbf S=\mathbf y$ instead of $|\mathbf S-\mathbf y|<1$ in the definition of $N$". Reduction to the page's question (authored, detailed on the result page): for $A=\{a_1<\cdots<a_N\}$, the subsets with $|S|=l$ and $\sum S=t$ are the sign vectors with $\sum_k\varepsilon_k(a_k,1)=(2t-\sum A,\,2l-N)$, at most the number of sums in the unit ball around that point; the vectors $(a_k,1)$ are $1$-separated since the $a_k$ are distinct integers; the condition of Theorem 1 fails for $a_k=k$ and $\mathbf e=(-1/n,\sqrt{1-n^{-2}})$, where every $|(\mathbf a_k,\mathbf e)|<1$, but the fixed-size count is unchanged by translating $A$ (the sum shifts by $c\sum_k\varepsilon_k=c(2l-N)$), and after moving the middle element to $0$ at least $\lfloor N/2\rfloor-1\ge N/4$ of the $a_k$ satisfy $|(\mathbf a_k,\mathbf e)|\ge1$ for every unit $\mathbf e$ once $N\ge6$ (for $\mathbf e=(\cos\theta,\sin\theta)$ with $\sin\theta\ge0$, those with $a_k\cos\theta\ge0$ and $|a_k|\ge1$, since $|\cos\theta|\ge\cos^2\theta\ge1-\sin\theta$). So the count is at most $c(1/4,2)2^N/N^2$ for $N\ge6$, and at most $2^N\le25\cdot2^N/N^2$ for $N\le5$: the second question's bound with an absolute constant, independent of $l$, $t$ and $A$. That the remark does not spell out the condition of Theorem 1 for the vectors $(a_k,1)$ is a filing observation, not a review verdict. Read depth: claims checked for Theorems 1, 2 and 4 and the remark; the proof of Theorem 2 (p. 208) is a paragraph modifying the proof of Theorem 4 (§ 3, pp. 200--208: Esséen's inequality for the concentration function, $|\varphi(\mathbf t)|\le\exp\{-f(\mathbf t)/2\}$ for the characteristic function of the sum, and measure bounds for the level sets of $f$), removing the mass at $\mathbf 0$ of the symmetrized distribution and using that a unit ball holds a bounded number of the $\mathbf a_k$ so that the paper's $\mu$ is bounded, "and this accounts for the gain of $n^{-1}$"; that paragraph was followed, the proof of Theorem 4 was read for structure only, and nothing was checked. Acceptance: refereed publication in Periodica Mathematica Hungarica (received 29 January 1976; Crossref record); the site's attribution; the conjecture is Erdős's, stated in 1965 with the constant independent of the subset size (his $t$, the page's $l$) and in 1973, as (8.8), with the constant independent of $t$, $l$, $n$ and the sequence, and reported there as never proved, before the paper was received.

Search scope. None of the routes below found a dispute of the theorems or a Lean file for the problem.

• The site: problem page, discussion thread and proof-claim tab; the full directory listing of formal-conjectures at its revision of 2026-09-18 (no file for the problem); the community database at its 2026-09-18 revision.
• The primary sources: [SaSz65] pp. 205--208; [St80] pp. 168, 178--179 and 184; [Er65] pp. 183--186, [Er73] p. 129 and [ErGr80] p. 59; [Ha77] pp. 197--211.
• Crossref: the DOI records 10.4064/aa-11-2-205-208, 10.1137/0601021 and 10.1007/BF02018403, and a bibliographic query for the Acta Arithmetica paper.
• Semantic Scholar: the citing records of the Sárközy--Szemerédi paper (65, mostly anticoncentration and Littlewood--Offord literature, among them [Ha77] and [Ng12]; titles read) and of Stanley's paper (500, Sperner and anticoncentration literature; titles scanned for Erdős--Moser items).
• arXiv: the record of 1112.0755.

Not searched: MathSciNet, zbMATH, Google Scholar, X, the site's reference service. Not held: van Lint 1967, Peck, Katona.

Remaining gaps. (1) The second question rests on Halász's Theorem 2, whose printed proof is a one-paragraph modification (p. 208) of the proof of Theorem 4, and on the authored reduction from the page's fixed-size subset counts to his unit-ball counts of signed sums in the plane; the paper's remark is the nearest printed statement of the page's inequality and does not spell out the condition of Theorem 1 for the vectors $(a_k,1)$. (2) The Lean development is neither built nor audited here. (3) Proof coverage: claims checked throughout; the Sárközy--Szemerédi proof and Halász's proof of Theorem 4 were read for structure only, Stanley's proofs and the Erdős--Moser Theorem 1 were not checked, and nothing is independently reviewed. (4) The exact maximum for positive sets, the middle coefficient of $\prod_{i\le N}(1+q^i)$, is identified by Corollary 5.1, but its asymptotic constant is not compiled here.

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_1965_extremal_problems_number_theory
• erdos_1973_problems_results_combinatorial_number_theory
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• halasz_1977_estimates_concentration_function_combinatorial_number_theory_probability
• halasz_1977_estimates_concentration_function_combinatorial_number_theory_probability / theorem_2
• halasz_1977_estimates_concentration_function_combinatorial_number_theory_probability / theorem_4
• sarkozi_1965_uber_ein_problem_von_erdos_und
• sarkozi_1965_uber_ein_problem_von_erdos_und / satz
• stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner
• stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner / corollary_5_1
• stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner / corollary_5_3
• stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner / theorem_2_4
• stanley_1980_weyl_groups_hard_lefschetz_theorem_sperner / theorem_3_1