Problem 873

Erdős Problem #873: declared status 'open'. Formalized: yes. Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that$$[a_i,a_{i+1},\ldots,a_{i+k-1}] < X,$$where the left-hand side is the least common multiple. Is it true that, for every $\epsilon >0$, there exists some $k$ such that$$F(A,X,k)<X^\epsilon?$$ Prize: no. Tags: number theory.