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Statement

Here $A_x$ is, in the paper's words (p. 315), "a set of positive integers with the least common multiple of each pair of terms not exceeding $x$ and $|A_x|$ being the largest". Theorem (pp. 315--316): "Let $x$ be a large real number. Then $|A_x|=\sqrt{\frac98x}+R(x)$, where $-2\leq R(x)\leq45\sqrt{\frac{x}{\log x}}\log\log x$."

The factor $\log\log x$ stands outside the square root. Remark (p. 316): the constant $45$ can be improved. Conjecture (p. 316): $R(x)\to\infty$ as $x\to\infty$.

Source. Li-Xia Dai and Yong-Gao Chen, Sequences with bounded l.c.m. of each pair of terms II, Acta Arith. 124 (2006), no. 4, 315--326, DOI 10.4064/aa124-4-2 (Crossref record read); the Theorem on printed pp. 315--316 = PDF pp. 1--2 of the publisher's 12-page file, read in the text layer and on the page images.

Read depth. Claims checked: the Theorem, the Remark and the Conjecture were read clause by clause on the page images of pp. 315--316. The proof (Sections 2--3, pp. 316--326) was not read beyond the statement of Lemma 1.

Proof pointer

Lemma 1 (p. 316) is Brun's pure sieve in the form of Halberstam--Richert, Lemma 2 the Rosser--Schoenfeld estimates for $\sum_{p\le z}1/p$; Section 3 applies them to the elements of $A_x$ outside Erdős's construction $B_x$. Not read here.

Dependencies

Brun's pure sieve; Rosser--Schoenfeld 1962 (the paper's [8], [9]).

Bears on

• Problem 441: the quantitative form of the first answer, $g(N)\le(9N/8)^{1/2}+O((N/\log N)^{1/2}\log\log N)$ as the site quotes it, with the explicit constant $45$ and the lower bound $g(N)\ge(9N/8)^{1/2}-2$.