Question 34
Suppose $a_{1}, a_{2}, \ldots, a_{8}$ are eight distinct integers from $\{1,2, \ldots, 16,17\}$. Show that there is an integer $k\gt 0$ such that the equation $a_{i}-a_{j}=k$ has at least three different solutions. Also, find a specific set of 7 distinct integers from $\{1,2, \ldots, 16,17\}$ such that the equation $a_{i}-a_{j}=k$ does not have three distinct solutions for any $k\gt 0$.