Solved. The label is the site's (SOLVED, page last edited 5 April 2026); its commentary states that the statement fails already at $k=3$. The answer is no: the commentary credits Furstenberg [Fu81] with an infinite set $B$ whose difference set $B-B$ is not $2$-intersective, which gives, for some fixed $\delta>0$ and every $m$, infinitely many $N$ with a set $A\subseteq\{1,\ldots,N\}$ of at least $\delta N$ elements and an $m$-element $B$ such that no $3$-term progression in $A$ has its difference in $B-B$. The standing is derived from the claim page: the accepted claim is Furstenberg's example and the site's deduction from it, on its claim page (Furstenberg, 1981), accepted on the curator's credit; a $k$-term progression contains a $3$-term one with the same difference, so the statement fails, at that $\delta$, for every $k\ge3$. Erdős attributes the question to himself and Mauldin, motivated by a problem in measure theory.