Problem 466

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claims/: The 1 claim page of Problem 466, one per claimant's result; the problem's standing derives from them.

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Statement. Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that

$$\| \lvert P_i-P_j\rvert \| \geq \delta$$

for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that

$$\lim_{x\to \infty}N(X,\delta)=\infty?$$

Statement (corrected). Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that

$$\| \lvert P_i-P_j\rvert \| \geq \delta$$

for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that

$$\lim_{X\to \infty}N(X,\delta)=\infty?$$

Notes. The site's display takes its limit in a variable $x$ that does not occur in $N(X,\delta)$, so read literally the display does not take the limit in the radius, and a comment in the site's discussion thread (11 May 2026) asks for the capital letter. The change replaces the lowercase $x$ under the limit by the radius $X$, the only variable the expression contains; nothing else changes. The evidence is Erdős's own statement of the question in his 1982 survey [Er82e], Chapter II, §9, printed p. 67, which writes the radius as $x$ throughout and conjectures "$N(x,\delta)\to\infty$", the limit in the radius; Sárközy, who states the question as Erdős's conjecture, writes "$\lim_{X\to+\infty}N(X,\delta_0)=+\infty$" in Part II of [Sa76], printed p. 105, and as (4) in Part I, printed p. 37. The defect is not the site's alone: the 1980 monograph [ErGr80], printed p. 93, prints "$\lim_{x\to\infty}N(X,\delta_0)=\infty$" with the lowercase $x$ the site reproduces, so the misprint is the monograph's and the site's wording copies it. The site's commentary, which credits Graham and Sárközy with proving the question, answers the corrected statement. The standing judges the corrected Statement.

Formulation. The site's wording, accessed 2026-09-18 (page last edited 16 September 2025; the thread's correction is not applied). As on Problem 465, the points lie in the disc of radius $X$, the condition forces $\delta\le1/2$ once there are two points, and $N(X,\delta)$ is nondecreasing in $X$ (a disc of radius $X$ contains every smaller disc), so "$N(X,\delta)\to\infty$ as $X\to\infty$" and "$N(X,\delta)$ is unbounded in $X$" say the same thing (an authored one-line remark). Erdős's 1982 survey writes the radius as a lowercase $x$ throughout and Sárközy as a capital $X$; both state the question with the limit in the radius, and the corrected Statement keeps the site's capital $X$.

Status. PROVED, the site's label (page last edited 16 September 2025), which describes the corrected Statement: the commentary credits Graham's construction and Sárközy's power lower bound. The result is recorded as an accepted full claim: Sárközy's Theorem 1 of [Sa76] Part II (Studia Sci. Math. Hungar. 11 (1976), refereed; p. 106) gives $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for $0<\delta\le1/(6\cdot8^4)$ and $X$ large depending on $\delta$, so $N(X,\delta)\to\infty$ as $X\to\infty$ for every such $\delta$ and in particular for $\delta=1/(6\cdot8^4)$; the same paper (pp. 105--106) reports Graham's construction, the points $(x_i,x_i^2)$ with $x_1=10$, $x_n=2x_{n-1}$, which gives $N(X,1/10)>\frac1{10}\log X$ for large $X$, the result the site's commentary credits to Graham. Graham's argument is reported by Sárközy with a sketch, while Erdős (1980, 1982) reports Graham's bound without a construction, and no separate paper of Graham's was located, so the claim page rests first-hand on Sárközy's Theorem 1 and reports Graham's construction second-hand, as Sárközy gives it, without depending on it. The derived standing is solved, proved.

Source. erdosproblems.com/466, accessed 2026-09-18: the problem page (PROVED, with the site's note that the answer is affirmative; last edited 16 September 2025; source keys [Er72], [ErGr80], [Er82e]; commentary citing [Sa76] and Problems 465 and 953; "Formalised statement? No"), its one-comment discussion thread (11 May 2026) and its empty proof-claims tab. Cite as: T. F. Bloom, Erdős Problem #466, https://www.erdosproblems.com/466, accessed 2026-09-18.

References.

• [Sa76] Sárközy, A., On distances near integers. I, II. Studia Sci. Math. Hungar. 11 (1976), 37--50 (received 10 December 1975) and 105--111 (received 11 February 1976). Part II: Graham's construction, pp. 105--106; Theorem 1, p. 106; Corollary, p. 107; the remark $N(X,1/10)>X^C$, p. 110. Part I: conjecture (4) and Graham's proof of it attested, p. 37. Library homes: sarkozy_1976_distances_near_integers_ii and sarkozy_1976_distances_near_integers_i (both in the REAL-J open scan of the whole 1976 volume).
• [Er72] Erdős, P., Extremal problems in number theory. Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, 1972), 80--86. Section IV, printed p. 83. Library home: erdos_1972_extremal_problems_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 pp. 92--93. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [Er82e] Erdős, P., Some of my favourite problems which recently have been solved. Proceedings of the International Mathematical Conference (Singapore, 1981), North-Holland Math. Stud. 74 (1982), 59--79. Chapter II, §9, printed pp. 67--68. Library home: erdos_1982_my_favourite_problems_which_recently_have.
• [Ko01] Konyagin, S. V., On the distances between points on the plane. Mat. Zametki 69 (2001), no. 4, 630--633 (in Russian; Math. Notes 69 (2001), 578--581). Its introduction (p. 630) attests Graham's proof and Sárközy's lower bounds; its Theorem is the upper bound of Problem 465. Library home: konyagin_2001_distances_between_points_plane.
• [GoMo26] Goenka, R. and Moore, K., Point sets avoiding near-integer distances. arXiv:2605.06621v1 (7 May 2026). A lead on the higher-dimensional analog; only its abstract was read.

Formalization. A statement and a third-party proof, neither built nor audited in this corpus. The file ErdosProblems/466.lean of formal-conjectures, added on 20 September 2026, declares erdos_466 : answer(True) ↔ ∃ δ : ℝ, 0 < δ ∧ Tendsto (fun X ↦ N X δ) atTop atTop under category research solved, which states the corrected Statement, with the radius as the limit variable. Its formal_proof attribute names Erdos466.lean in Boris Alexeev's repository plby/lean-proofs, whose header names Ronald Graham as the informal author and the AI systems Codex and GPT-5.6 Sol as the formal authors; it proves $N(X,1/10)\to\infty$ from the points $(2^m,4^m)$. The variants erdos_466.variants.graham ($\log X/10<N(X,1/10)$ for $X\ge1$) and erdos_466.variants.sarkozy (Sárközy's bound) are stated with proof sorry. The community database records the statement formalized since 20 September 2026.

Current assessment

The question (site formulation). The statement above; PROVED. The commentary credits Graham with the answer yes, with the bound $N(X,1/10)>\frac{\log X}{10}$, and Sárközy [Sa76] with the much stronger $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for all sufficiently small $\delta>0$, and points to Problem 465 for upper bounds and to Problem 953 for a related problem. The one comment (11 May 2026) asks that the $x$ under the limit be a capital letter and supplies page numbers for the three source keys (p. 83 of [Er72], p. 92 of [ErGr80], p. 67 of [Er82e]). The proof-claims tab is empty. The community database records the problem proved since 31 August 2025 and the statement formalized since 20 September 2026.

Erdős's statements. [ErGr80], printed pp. 92--93: after defining $N(X,\delta)$ as on this page, "Erdös conjectured that for any $\delta\in(0,1/2)$, $N(X,\delta)=o(X)$, and, on the other hand, there is a $\delta_0>0$ so that $\lim_{x\to\infty}N(X,\delta_0)=\infty$." The passage then reports that Graham proved the second conjecture with $N(X,1/10)>\frac1{10}\log X$ (cited as [Gr ($\infty$)]) and that Sárközy improved this to $N(X,1/10)>X^c$ for an absolute constant $c$ and, for every $\varepsilon>0$, to $N(X,\delta)>X^{1/2-\varepsilon}$ for $\delta\le\delta(\varepsilon)$ and large $X$ (cited as [Sár (xx) b]). The lowercase $x$ under the limit is the monograph's; the passage is compiled on Problem 465. [Er82e], Chapter II, §9, printed pp. 67--68: "I conjectured that $N(x,\delta)=o(x)$ and $N(x,\delta)\to\infty$. [...] Graham proved the second conjecture, he in fact proved $N(x,1/10)>\frac1{10}\log x$. Sárközy showed that to every $\varepsilon>0$ there is a $\delta(\varepsilon)>0$ so that for every $\delta<\delta(\varepsilon)$ $N(x,\delta)>x^{1/2-\varepsilon}$." [Er72], Section IV, printed p. 83, for complex $z_i$ with $|z_i|<n$ whose mutual distances differ from every integer by more than $c$: "Graham and Sárközi showed that for every $c$ $(0<c<\frac12)$ $t>n^{\alpha_c}$ $\alpha_c$ [sic] $(\alpha_c<\frac12)$, and Sárközi proved $t<c\,n/\log\log n$" (the stray $\alpha_c$ is the print's). In 1972 Erdős thus credited a power lower bound for every $c$ to Graham and Sárközy jointly, four years before Sárközy's papers appeared; the 1976 and 1980 accounts credit Graham with the logarithmic bound and Sárközy with the power bound. Recorded as printed.

Status support. Sárközy's Part II (Theorem 1, p. 106): "Let $0<\delta\le\frac1{6\cdot8^4}$ (5) and $X$ be sufficiently large depending on $\delta$. Then $N(X,\delta)>X^{1/2-\delta^{1/7}}$ (6)", with the Corollary (p. 107) that for every $\varepsilon>0$ there is $\delta_0(\varepsilon)>0$ with $N(X,\delta)>X^{1/2-\varepsilon}$ for $0<\delta<\delta_0(\varepsilon)$ and $X$ large. Since the exponent is positive in the range (5), $N(X,\delta)\to\infty$ for every such $\delta$, which proves the corrected Statement. The paper's p. 105 reports the question as Erdős's conjecture ("Erdős conjectured that for some $\delta_0>0$, $\lim_{X\to+\infty}N(X,\delta_0)=+\infty$. This conjecture has been proved by R. L. Graham (Erdős's oral communication)") and gives Graham's construction: "let $x_1=10$ and $x_n=2x_{n-1}$ for $n=2,3,\ldots$, and let us define the positive integer $m$ by $\sqrt{x_m^2+x_m^4}\le X<\sqrt{x_{m+1}^2+x_m^4}$. For $i=1,2,\ldots,m$, let $P_i$ denote that point whose Cartesian coordinates are $x=x_i$, $y=x_i^2$: $P_i=(x_i,x_i^2)$. These points are in the circle of radius $X$ (with center at the origin) and it is easy to show that for large enough $X$, $m>\frac1{10}\log X$ and $\varrho(P_i,P_j)>\frac1{10}$ for $1\le i<j\le m$" (pp. 105--106; the last root on p. 105 is printed with $x_m^4$ where $x_{m+1}^4$ is meant, and the last inequality without the double bars of the norm, whose presence the conclusion requires), "Hence, (2) $N(X,1/10)>\frac1{10}\log X$ for large enough $X$." The proof of Theorem 1 (pp. 107--110) was read for structure only: the points $(\sum_{i=0}^t\varepsilon_ik^{2i+2},\sum_{i=0}^t\varepsilon_ik^i)$ with digits $0\le\varepsilon_i\le k-2$, $k>\delta^{-1/7}$, of which there are $(k-1)^{t+1}>X^{1/2-\delta^{1/7}}$ inside the disc, and the Lemma of p. 106 (if $a$ is a positive integer and $3\delta<b^2/a<2(1-\delta)$ then $\|\sqrt{a^2+b^2}\|>\delta$) applied to the coordinate differences. The remark on p. 110 adds that the same method gives $N(X,1/10)>X^C$ for an absolute constant $C$. Acceptance: a refereed journal (Studia Sci. Math. Hungar.); the monograph's [Sár (xx) b] and Konyagin's [2] cite it; the site accepts it. Part I (p. 37) records the conjecture as (4) and attests Graham's proof from Erdős's oral communication; Konyagin's introduction (p. 630) attests that R. L. Graham proved the conjecture and reports Sárközy's two lower bounds. Graham's construction has no separate publication among the sources read: the monograph cites "[Gr ($\infty$)]", Sárközy alone gives the construction, with the words "it is easy to show", Erdős (1980, 1982) reports Graham's bound without a construction, and the site names no paper of Graham's; the accepted claim does not depend on it.

The upper bounds and the gap. Problem 465's page compiles the upper bounds: Sárközy's $N(X,\delta)<(4\cdot10^4/\delta^3)X/\log\log X$ (Part I, p. 38) and Konyagin's $N(X,\delta)<C(\delta)X^{1/2}$ for $X\ge1$ (Konyagin's Theorem). Together with Theorem 1 above, $X^{1/2-\varepsilon}<N(X,\delta)<C(\delta)X^{1/2}$ for $\delta<\delta_0(\varepsilon)$ and large $X$; for a fixed $\delta$ the exact order is not known from the sources read, and for $\delta=1/10$ the known lower bound is $X^C$ with an unspecified absolute $C$. The higher-dimensional analog is a lead ([GoMo26], abstract only).

Search scope. The routes of Problem 465's search were run for both pages: the site (problem page, thread and tab), the formal-conjectures listing and the community database; the primary sources [Sa76] I--II, [ErGr80], [Er82e], [Er72] and [Ko01] at the pages stated; the mathnet.ru and Crossref records of [Ko01]; Semantic Scholar's record of the Math. Notes translation (no citing records; its title search was unavailable and is not covered); the arXiv API query all:"near integers" AND all:distances (three records, one of them [GoMo26]) and the two lead abstracts; REAL-J's volume list and the 1976 volume, the open scan that contains the two papers. Not searched: MathSciNet, zbMATH, Google Scholar, X. Nothing found disputes the results or names a paper of Graham's.

Remaining gaps. (1) The site's wording keeps the misprinted limit variable that the thread asks to correct; the standing judges the corrected Statement. (2) Graham's construction is known only as Sárközy reports it, with a sketch, and as the third-party Lean proof under Formalization renders it, since Erdős reports the bound alone; the accepted claim rests on Sárközy's Theorem 1 first-hand. (3) Proof coverage: claims checked for Theorem 1 and its Corollary; the proof was read for structure only; nothing is independently reviewed. (4) The exact order of $N(X,\delta)$ for fixed $\delta$ is open (Problem 465). (5) The monograph card records the [ErGr80] passages for this page and Problem 465; Problem 953 is outside this page.

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_1978_set_theoretic
• erdos_1982_my_favourite_problems_which_recently_have
• chojecki_2026_order_growth_planar_sets_avoiding_integer
• chojecki_2026_order_growth_planar_sets_avoiding_integer / theorem_3
• erdos_1972_extremal_problems_number_theory
• erdos_1980_old_new_problems_results_combinatorial_number_theory
• konyagin_2001_distances_between_points_plane
• konyagin_2001_distances_between_points_plane / theorem
• sarkozy_1976_distances_near_integers_i
• sarkozy_1976_distances_near_integers_ii
• sarkozy_1976_distances_near_integers_ii / theorem_1