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Claim. Theorem 3 a) of K. Gyarmati, On divisibility properties of integers of the form $ab+1$, Period. Math. Hungar. 43 (2001), no. 1--2, 71--79, states that there is a set $\mathcal A\subseteq\{1,2,\dots,N\}$ with $|\mathcal A|\gg\log N$ and $a+a'$ squarefree for all $a,a'\in\mathcal A$. So $f(N)\gg\log N$ for the $f$ of Problem 1109. The paper says that Theorem 3 a) is due to Erdős and Sárközy and gives another proof, based, like its other lower bounds, on graph theory. The paper's main subject is the multiplicative analogue, sets with $ab+1$ squarefree; its Theorem 2 a) also gives sets $\mathcal A,\mathcal B\subseteq\{1,\dots,N\}$ with $|\mathcal A|=|\mathcal B|\gg(\log N)^2$ and $a+b$ squarefree for all $a\in\mathcal A$, $b\in\mathcal B$, a bound for the two-set variant, not for $f(N)$. The preprint link is the author's copy. The DOI record dates the issue August 2002, while the volume and the author's publication list give 2001; the page is named by the first day of the record's month.

Covers. The lower bound $f(N)\gg\log N$, by a second proof; the bound is Erdős and Sárközy's and is improved by Konyagin 2004. Neither question of the problem is answered.

Depends on. No page of this wiki.

Acceptance. Refereed: Period. Math. Hungar. 43 (2001), no. 1--2, 71--79. The site labels the problem OPEN, so its commentary crediting the proof is not reviewed evidence. The proof is not checked in this corpus.