Solved. Both questions are answered in Erdős and Szemerédi's 1980 paper (Mat. Lapok 28, 121--124, in Hungarian). Theorem I gives $\limsup A(x)/x^{1/2}\le c$ with $c=\sum_{k\ge1}(k^{1/2}-(k-1)^{1/2})/k=1.8600\ldots$, so $A(x)\le(c+o(1))x^{1/2}$ and the answer to the first question is yes; Theorem II gives $\liminf A(x)/x^{1/2}\le1$ for every $A$, and $A=\mathbb N$ attains $1$, so the largest possible value of the liminf is exactly $1$. The printed proof of Theorem II ends with a false numerical assertion ($\gamma<1$ for a series equal to $1.1840\ldots$) and does not close as printed; the theorem is true, by the authored averaging proof under Current assessment, which is a note of this corpus and not acceptance evidence. Mat. Lapok is the refereed journal of the Bolyai Society (the card records MR 82c:10066 and Zbl 476.10045). Claim page: Erdős and Szemerédi 1980 (accepted; refereed, and credited by the site's curator), which also links the public Lean file of August 2026 that declares itself a formalization of their result (not built or audited in this corpus).