Question 4
Let $n$ and $k$ be positive integers with $n\ge k\ge2$. For $i=1,\ldots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\ldots,n$, contains at least one integer in $X$.