Chen 1998 sequences bounded l c m each

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theorem: The asymptotic size of a largest set of positive integers whose pairwise least common multiples are at most x, with the extremal set differing from the natural construction by o(x^{1/2}) elements.

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Chen, Yong-Gao, Sequences with bounded l.c.m. of each pair of terms. Acta Arith. (1998), 71-95.

The paper is Acta Arithmetica 84 (1998), no. 1, 71--95, DOI 10.4064/aa-84-1-71-95 (Crossref record read). The retained folder-name PDF is the publisher's 25-page file; printed p. $n$ is PDF p. $n-70$, and the text layer is clean. Read status: claims checked for the Theorem and the Note that follows it, read clause by clause on the page images of pp. 71--72; the attribution of the bounds $1.638\sqrt x$ and $1.43\sqrt x$ to Choi (the paper's [1] and [2], Mathematika 19 (1972) and Acta Arith. 29 (1976)) was checked there too; the proof (pp. 72--95) was not read beyond the statement of Lemma 1. The file's text layer carries no copyright or license line; 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/aa-84-1-71-95, read 2026-10-02); the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

Let A_x be a largest set of positive integers whose pairwise least common multiples are all at most x; Erdos asked in 1951 for the size of A_x, with known bounds sqrt(9x/8) + O(1) <= |A_x| <= sqrt(4x) + O(1) and Choi's upper bounds 1.638 sqrt(x) and then 1.43 sqrt(x). The paper's Theorem determines the asymptotic: with B_x the union of the integers up to sqrt(x/2) and the even integers between sqrt(x/2) and sqrt(2x), it shows |A_x \ B_x| = o(sqrt(x)) and hence |A_x| = |B_x| + o(sqrt(x)) = sqrt(9x/8) + o(sqrt(x)), so the extremal set is essentially the natural construction B_x. The Note after the Theorem records |A_x cap B_x| = sqrt(9x/8) + o(sqrt(x)) and |B_x \ A_x| = o(sqrt(x)), and says the error term can be made explicit from the proof. The argument runs through a chain of sieve lemmas: Lemma 1 uses the Eratosthenes-Legendre sieve to find an integer k in a short interval near c_1 x^(1/2) all of whose shifted values a_i k + b_i have only large prime factors. This resolves the asymptotic form of Erdos problem 441 on the maximal size of such a sequence.

Source: https://matwbn.icm.edu.pl/ksiazki/aa/aa84/aa8416.pdf.

Bears on. #441

Results to transcribe.

• Theorem (p. 71): |A_x \ B_x| = o(sqrt(x)), hence |A_x| = |B_x| + o(sqrt(x)) = sqrt(9x/8) + o(sqrt(x)).
• Note after the Theorem (pp. 71--72): |A_x cap B_x| = sqrt(9x/8) + o(sqrt(x)) and |B_x \ A_x| = o(sqrt(x)), so A_x and B_x nearly coincide.
• Lemma 1 (p. 72): A sieve lemma producing k in (c_1 x^(1/2)+c_2, c_1(x^(1/2)+x^(1/4))+c_2) with every prime factor of the product of a_i k + b_i exceeding (log x)/(6 log M).