# Maximum number of infinite areas in the plane divided by $n$ lines

I know that when $$n$$ lines are drawn in the plane, the maximum possible number of regions the plane is divided into is $$\frac{n(n+1)}2+1$$. This maximum is achieved when no two of the lines are parallel, and no three are collinear.

When the plane is divided into regions by $$n$$ lines, some of these regions will be bounded, and some will be infinite. What is the maximum possible number of infinite regions in such an arrangement?

• What you mean is that the number of regions into which $n$ lines, no three of which intersect at the same point, divide a plane is $\frac{n(n + 1)}{2} + 1$. – N. F. Taussig Oct 1 '19 at 9:56
• I am confused of "infinite areas". Can you explain more? – Isaac YIU Math Studio Oct 1 '19 at 9:58
• Wrong. n parallel lines divide a plain into n + 1 areas. – William Elliot Oct 1 '19 at 10:45
• You have to impose a condition such as 'no 3 lines are concurrent' or something like that. Else the number of lines depends on the configuration. – Sam Oct 1 '19 at 11:58
• Are you asking how many of the regions are unbounded? – saulspatz Oct 1 '19 at 12:10

Imagine taking a circle $$C$$ which is large enough such that (i) all of the intersection points between lines in the arrangement lie inside $$C$$ and (ii) every line intersects $$C$$ twice.
It follows from (i) that bounded regions do not intersect $$C$$ (since all their vertices are inside $$C$$). Furthermore, any unbounded region must intersect $$C$$ (The region must have points outside $$C$$ since it's unbounded. Take a point on the boundary of the region lying outside $$C$$, follow the boundary line towards $$C$$ -- it can't intersect another boundary before hitting $$C$$ by (i))
Now imagine making a single circuit travelling around $$C$$. There will be $$2n$$ distinct points at which $$C$$ intersects one of the lines (since each line intersects twice and (i) implies that no two lines intersect $$C$$ in the same place). So we only change regions $$2n$$ times before returning back where we started, implying we've passed through at most $$2n$$ regions total. So the number of unbounded regions is at most $$2n$$.
Equality holds so long as we don't encounter the same region twice as we travel around the circle (which I believe only happens if we have $$n$$ parallel lines). For example, we get equality if we just take $$n$$ concurrent lines. The number of regions is $$2n$$ (very far from maximal), but they're all unbounded.