When colouring a map with known regions, a maximum of four colours are required.
How many colours are required when the regions are unknown?
The use-case I have in mind is in puzzles where you construct the regions as the puzzle progresses. Given that you don't know how many regions there are, and how they're arranged, how many colours do you need to use to guarantee connecting regions are different colours?
Two dimensional maps only.
Edits:
Newly discovered regions may be isolated from previously discovered regions.
Newly discovered regions may grow as new clues are discovered. Example of such a puzzle: https://youtu.be/emH5qSvPTes
Regions are not always constrained to be the same size
Regions are often constructed on an orthoganal grid, so no infinitesimals.
The grid is almost always finite, but periodic boundary conditions sometimes apply (eg https://youtu.be/xRoCWkm_8KE), although this is a special case.