Erdős Problem #36 — A translate matching $n$ pairs of a set and its complement

A somewhat similar question is the following one: Let $a_1, a_2, \ldots, a_{2n}$ be $2n$ integers in the interval $(1, 4n)$ and let $b_1, b_2, \ldots, b_{2n}$ be the other $2n$ integers of the same interval. Does there exist an integer $x$ so that the number of solutions of $a_i + x = b_j$ is at least $n$? If the $a$-s are the integers $n+1, n+2, \ldots, 3n$ we see immediately that the value $n$, if true is certainly best possible.