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Claim. Let $f(N)$ be the largest size of a set $A\subseteq\{1,\dots,N\}$ with $a+a'$ squarefree for all $a,a'\in A$, the case $a=a'$ included; S. V. Konyagin, Problems on the set of squarefree numbers, Izv. Math. 68 (2004), no. 3, 493--520 (English translation of Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 3, 63--90), calls this quantity $\mathrm{ES}_N$. Theorem 1 (1.4) and Theorem 3 (1.6) of the paper state that there are effective positive constants $c_1$ and $C_2$ with

$$c_1\log^2N\log\log N\le f(N)\le N^{11/15}\exp\left(C_2\frac{\log N}{\sqrt{\log\log N}}\right)$$

for all $N\ge3$. The upper bound comes from a large sieve inequality for square moduli (Theorem 2), the lower bound from Brun's sieve with quadratic moduli (Section 4) and a balanced sifting argument. The paper also records the expectation $f(N)=o(N^\varepsilon)$ for every $\varepsilon>0$, the first question of Problem 1109, as unproved. The source card is Konyagin 2004; the page name's date is the issue date in the DOI record.

Covers. The best known estimates of $f(N)$: $\log^2N\log\log N\ll f(N)\ll N^{11/15+o(1)}$, improving both bounds of Erdős and Sárközy 1987. Neither question of the problem is answered.

Depends on. No page of this wiki.

Acceptance. Refereed: Izv. Math. 68 (2004), no. 3, 493--520. The site labels the problem OPEN, so its commentary crediting the bounds is not reviewed evidence. The proof is not checked in this corpus.