Open. The site labels the problem OPEN (page last edited 3 December 2025). Its commentary credits three sets of bounds, each recorded on a claim page: Erdős and Sárközy's $\log N\ll f(N)\ll N^{3/4}\log N$, Erdős and Sárközy 1987; Gyarmati's second proof of $f(N)\gg\log N$, Gyarmati 2001; and Konyagin's $\log^2N\log\log N\ll f(N)\ll N^{11/15+o(1)}$, the best known, Konyagin 2004. None answers either question. G. N. Sárközy [Sa92c] extends the problem to sums $A+B$ and to $k$-power-free sums and, as Konyagin records (p. 494), improves the upper bound to $f(N)\ll N^{3/4}$; that bound is superseded by Konyagin's and the site does not credit it, so it has no claim page.