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Claim. Let $\mathrm{ES}_N$ 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. Theorem 1 (1.4) of 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), states that there is an effective constant $C_2$ with

$$\mathrm{ES}_N\le N^{11/15}\exp\left(C_2\frac{\log N}{\sqrt{\log\log N}}\right)$$

for all $N\ge3$. If $A=\{a_1<a_2<\cdots\}$ is an infinite set with squarefree sums, then $A\cap\{1,\dots,N\}$ is one of the sets counted by $\mathrm{ES}_N$, so inverting the bound gives

$$a_j\gg j^{15/11}\exp\left(-O\left(\frac{\log j}{\sqrt{\log\log j}}\right)\right),$$

that is $a_j\ge j^{15/11-o(1)}$, for all large $j$. The paper treats the finite problem; the inversion is stated on p. 4 of the second arXiv version of van Doorn and Tao, arXiv:2512.01087v2. The source card is Konyagin 2004; the page name's date is the issue date in the DOI record.

Covers. A necessary growth for Problem 1103: every infinite set with squarefree sums satisfies $a_j\ge j^{15/11-o(1)}$, the best lower bound known, which the site's commentary credits to Konyagin. The true growth rate is not determined.

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 bound is not reviewed evidence. The proof is not checked in this corpus.