Problem 708

Erdős Problem #708: declared status 'open'. Formalized: no. Let $g(n)$ be minimal such that for any $A\subseteq [2,\infty)\cap \mathbb{N}$ with $\lvert A\rvert =n$ and any set $I$ of $\max(A)$ consecutive integers there exists some $B\subseteq I$ with $\lvert B\rvert=g(n)$ such that$$\prod_{a\in A} a \mid \prod_{b\in B}b.$$Is it true that$$g(n) \leq (2+o(1))n?$$Or perhaps even $g(n)\leq 2n$? Current best: Their lower bound construction takes $A$ as the set of $p_ip_j$ for $i\neq j$, where $p_1<\cdots <p_\ell$ is some set of primes such that $2p_1^2>p_\ell^2$. (In 1992 1000 rupees was worth approximately \$38.60.) Erd\H{o}s and Sur\'{a}nyi similarly asked what is the smallest$c_n\geq 1$such that in any interval$I\subset [0,\infty)$of length$c_n\max(A)$there exists some$B\subseteq I\cap \mathbb{N}$with$\lvert B\rvert=n such that\[\prod_{a\in A} a \mid \prod_{b\in B}b.\]They prove c_2=1$and$c_3=\sqrt{2}$, but have no good upper or lower bounds in general. Prize:$100. Tags: number theory.