Problem 635

Erdős Problem #635: declared status 'open'. Formalized: no. Let $t\geq 1$ and $A\subseteq \{1,\ldots,N\}$ be such that whenever $a,b\in A$ with $b-a\geq t$ we have $b-a\nmid b$. How large can $\lvert A\rvert$ be? Is it true that$$\lvert A\rvert \leq \left(\frac{1}{2}+o_t(1)\right)N?$$ Prize: no. Tags: number theory.