Question 1
Due to the increasing pandemic situation, the government decides to enforce traffic ban by building $n$ straight lines (infinite at both directions) as "walls", with no two of them are parallel, and no three of them are concurrent. The set of these walls is denoted as $W_n$. For each wall in $W_n$, the government paints one of its sides in blue, and the other side in green. By such, for any two walls, their intersection is divided into four corners: a mono-color corner such that both walls are blue, a mono-color corner such that both walls are green, and two duel-color corners such that the walls are in different colors. To make sure there are paths for emergency, the government opens a door on each intersection of walls so that people can travel from a duel-color corner to the other duel-color corner. These doors are the only ways for people to cross a wall.
Given $W_n$, let $k(W_n)$ be the largest positive integer $k$ such that, no matter how the government paints the walls, we can place $k$ people on the plane so that no two of them can ever meet each other. For each $n$, find all possible values of $k(W_n)$.
Given $W_n$, let $k(W_n)$ be the largest positive integer $k$ such that, no matter how the government paints the walls, we can place $k$ people on the plane so that no two of them can ever meet each other. For each $n$, find all possible values of $k(W_n)$.