MathNet 2024-part-01 — 168 problems (EN–VI) Question 61

Let $n \ge 4$ be an integer number and $S_n = \{1, 2, 3, \dots, 2^n\}$. Two sets $A, B$ are given, $A \subset S_n, B \subset S_n \setminus S_{n-1}$, such that $|A| = n+1, |B| = 2$. Is it possible $ab-1$ be a perfect cube for any $a \in A, b \in B$?
(Dragomir Grozev)