How Many Colours Will You Need If Colouring Polygons Enclosed By Straight Lines? There is an infinite plane.
I can draw straight infinitely long lines in the plane(mind you,only finitely many).
Now,I want to colour the polygons.
The colours of adjacent polygons must not be the same(if the polygons only share the same point,then they aren't adjacent).
How many colours do I need to colour the polygons enclosed by the straight line(including the infinite ones)?
Example
I think the answer is two, but I couldn't prove it.
Can anyone prove it for me(or prove that it needs 3 or more colours)?

Two colours suffice.
Think of adding the lines one at a time, starting with the plane coloured, say, red.  Each time you add a line, leave one of the half-planes into which it divides the plane alone, and flip all the colours (red $\to$ blue, blue $\to$ red) in the other half-plane.  Thus any region cut by the new line becomes two regions, one red and one blue.
Any two adjacent regions will still be different colours.
