Question 137
Let $p$ be a given prime number and $n, k \ge 2$ be integers. For each integer $i = 1, 2, \dots, n$, the set $S_i$ is constructed by choosing exactly one square from each column of a given $p \times k$ chessboard. Given that $|S_i \cap S_j| = 1$ for all $1 \le i \lt j \le n$ and that for every pair of squares belonging to different columns there exists an $S_i$ containing these two squares, find all possible pairs $(n, k)$ in terms of $p$.