ILL 1992 Problem 39

Let $n \geq 2$ be an integer. Find the minimum $k$ for which there exists a partition of $\{1, 2, . . . , k\}$ into $n$ subsets $X_1,X_2, \cdots , X_n$ such that the following condition holds: for any $i, j, 1 \leq i < j \leq n$ , there exist $x_i \in X_1, x_j \in X_2$ such that $|x_i - x_j | = 1.$