Hudson outlines that for separation of the plane into $n$ contiguous coloured regions of finite interior but infinite boundary so that no two colours touch one requires $n$ colours in his article "Four Colors Do Not Suffice" 2003.
On the other hand the famous four colour theorem states that for separation of the plane into arbitrary contiguous coloured regions of finite interior and finite boundary a maximum of four colours is required.
What is the minimum number of colour required for an arbitrary map of $n$ countries with $a$ countries with finite interior and infinite boundary and $b$ countries with finite interior and finite boundary such that $a + b = n$?