The map shows the four colour theorem in practice.
… given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.
In actual, fact the map uses 5 colours, if you include the white used for the oceans, although it would still technically be possible to draw a world map using only 4 colours, if landlocked countries shared the same colour as the ocean.
The four-color theorem does not prove that the world map can be four-colored, because some countries consist of non-contiguous regions (like the U.S., Russia, and Azerbaijan). It happens to work out anyway due to the placement of those countries.