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Claim. For $0<\delta\le1/(6\cdot8^4)$ and $X$ large depending on $\delta$,

$$N(X,\delta)>X^{1/2-\delta^{1/7}},$$

where $N(X,\delta)$ is the largest number of points in a disc of radius $X$ whose pairwise distances all lie at distance at least $\delta$ from the nearest integer, as Problem 466 defines it. This is Theorem 1 of Part II (p. 106), quoted on the result page theorem_1, with its Corollary (p. 107): 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 large $X$. The digest is on the source card sarkozy_1976_distances_near_integers_ii. The exponent is positive throughout the range, so $N(X,\delta)\to\infty$ as $X\to\infty$ for every such $\delta$. The same paper (pp. 105--106) gives Graham's construction, the points $(x_i,x_i^2)$ with $x_1=10$ and $x_n=2x_{n-1}$, which yields $N(X,1/10)>\frac1{10}\log X$ for large $X$ and is the result the site's commentary credits to Graham. Graham's argument has no publication of its own among the sources read: Sárközy alone sketches it, from Erdős's oral communication, while Erdős reports Graham's bound without a construction in the 1980 monograph and the 1982 survey. It is disclosed here and gets no page. The proof of Theorem 1 (pp. 107--110: points with coordinates built from digits in base $k>\delta^{-1/7}$ and the Lemma on $\|\sqrt{a^2+b^2}\|$) is not checked here. The matching upper bounds are compiled on Problem 465.

Acceptance. Refereed: A. Sárközy, On distances near integers, II, Studia Sci. Math. Hungar. 11 (1976), 105--111, received 11 February 1976. Reviewed: the site's curator, Thomas F. Bloom, marks Problem 466 PROVED and credits Graham's proof and Sárközy's improvement in the problem's commentary (page last edited 16 September 2025); the curator neither wrote nor submitted the result. The 1980 monograph of Erdős and Graham and Konyagin's introduction of 2001 cite the bound as proved. Graham's construction has a third-party Lean formalization, linked above, whose header names Graham as the informal author and the AI systems Codex and GPT-5.6 Sol as the formal authors. It was not built here, so no formalized evidence is listed, and Theorem 1 itself is not formalized. The page is dated to the year of publication, since the volume prints no fuller date for the article; the paper link leads to the repository record of the volume scan.