Chen 2007 sequences bounded l c m each

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corollary_1: The remainder in the asymptotic for the largest set with pairwise least common multiple at most x is unbounded along an infinite sequence of x.

theorem_1: Among sets with pairwise least common multiple at most x that contain Erdős's construction B_x, a largest one equals B_x for infinitely many x and exceeds it by at least the iterated-logarithm count loc x minus two for infinitely many x.

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Chen, Yong-Gao and Dai, Li-Xia, Sequences with bounded l.c.m. of each pair of terms. {III}. Acta Arith. (2007), 125-133.

The paper is Acta Arithmetica 128 (2007), no. 2, 125--133, DOI 10.4064/aa128-2-3 (Crossref record read). The retained folder-name PDF is the publisher's 9-page file; printed p. $n$ is PDF p. $n-124$, and the text layer is clean. Read status: claims checked for Theorem 1 and Corollary 1, read clause by clause on the page image of p. 126; the Definition, Theorems 2 and 3, Corollary 2 and Problems 1 and 2 (pp. 126--127) were read in the text layer as statements; the proofs (pp. 127--133) were not read beyond the statement of Lemma 1. The file prints "© Instytut Matematyczny PAN, 2007" on its first page; the publisher's record labels the PDF download "Pobierz zgodnie z CC-BY", which the English site renders "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named (https://www.impan.pl/get/doi/10.4064/aa128-2-3, read 2026-10-02), and that named license on the publisher's page decides over the printed line; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Continuing Chen's asymptotic |A_x| = sqrt(9x/8) + o(sqrt(x)) for sets with pairwise least common multiples at most x, the paper studies the remainder terms R(x) = |A_x| - sqrt(9x/8) and R_1(x) = |C_x| - |B_x|, where C_x is a largest such set containing the standard construction B_x and Dai and Chen had shown -2 <= R(x) <= 45 sqrt(x/log x) log log x (the 2006 theorem; the 2007 introduction, p. 126, prints the upper bound with an extra sqrt(9x/8) term). Theorem 1 proves R_1(x) = 0 for infinitely many x but also R_1(x) >= loc x - 2 for infinitely many x, where loc x is the number of iterated logarithms needed to bring x below 1; Corollary 1 transfers the lower bound to R(x), so R(x) = O(1) fails. Theorems 2 and 3 give conditional upper bounds R_1(x) = O(log ... log x) with r+1 logarithms, and R_1(x) <= 2 loc x + O(1), under hypotheses about pairs of u-compromise integers (a notion introduced here in terms of primes dividing consecutive shifts of s and t); Corollary 2 states the unconditional R_1(x) = O(log log x). Problems 1 and 2 ask whether the u-compromise hypotheses hold. This sharpens the error term in Erdos problem 441 on the maximal size of sequences with bounded pairwise least common multiple.

Source: https://doi.org/10.4064/aa128-2-3.

Bears on. #441

Results to transcribe.

• Theorem 1 (p. 126): (i) R_1(x) = 0 for infinitely many x; (ii) R_1(x) >= loc x - 2 for infinitely many x.
• Corollary 1 (p. 126): R(x) >= loc x - 2 for infinitely many x, so the remainder is not bounded.
• Theorem 2 (pp. 126--127): Under a u-compromise separation hypothesis with r iterated logarithms and gap tau, R_1(x) = O(log...log x) with r+1 logarithms.
• Corollary 2 (p. 127): R_1(x) = O(log log x).
• Theorem 3 (p. 127): Under a halving hypothesis for u-compromise pairs (r iterated logarithms of t at least half of r-1 iterated logarithms of s), R_1(x) <= 2 loc x + O(1); the paper calls this hypothesis (its Problem 2) stronger than Theorem 2's (its Problem 1).