Hudson's paper: "Four Colours do not Suffice" 2003

Question

According to Wikipedia, for the Four Colour Theorem:

"bizarre regions, such as those with finite area but infinitely long perimeter, are not allowed; maps with such regions can require more than four colors"

citing a paper by Hudson of 2003, entitled "Four Colours do not Suffice".

I am unable to access this paper at present and wondered if anyone knew what conclusions and formulations it contains.

Thank you.

Answer 1

I don't have the paper, but there's a classic construction (the "island with two lakes") that builds three connected regions that share a common boundary.

From Croom's Basic Concepts in Algebraic Topology (Google Books link):

Consider the double annulus in Figure 1.1 as an island with two lakes having water of distinct colors surrounded by the ocean. By constructing canals from the ocean and the lakes into the island, we shall define three connected open sets. First, canals are constructed bringing water from the sea and from each lake to within distance $d=1$ of each dry point of the island. This process is repeated for $d=\frac12, \frac14, \ldots, (\frac12)^n, \ldots$, with no intersection of canals. The two lakes with their canal systems and the ocean with its canal form three regions in the plane with the remaining "dry land" $D$ as common boundary. Since $D$ separates the plane into three connected open sets instead of two, the Jordan Curve Theorem shows that $D$ is not a simple closed curve.

It can easily be extended to four, or five, or fifty, which shows that if you let your countries be arbitrary connected regions of the plane, you may need more than four colors to make a map.