The four color map theorem:

https://en.wikipedia.org/wiki/Four_color_theorem

states that for a 2D area broken up into irregularly shaped borders, only four different colors need to shade in the borders.

It’s coloured using 5 colours in the picture, but that’s just a random logo and has nothing to do with any theorem. You can colour it using 4 colours.

Red and blue aren’t in contact with each other, so they don’t need to be different colors.

Why? If the green part is yellow you’re fine; the green and yellow don’t touch each other. I’m not sure that picture works well anyway since it’s not a map. Though I guess if the blank spaces between the shapes are bodies of water you could theoretically have a map that looks like that.

ETA: Ninja’d but there are two ways to make 4 colors work there.

Yellow and green can be the same colour and red and blue can be the same colour. That leaves one colour for the background.

Two unconnected black regions, two unconnected blue regions, 4 each unconnected green and yellow regions.

(background assumed black)

That’s well done.

Hrumph.

Never challenge a math theorem without thinking it through twice, and then one more time.

'Tis the season for making a proof, checking it twice.

Indeed, verifying a proof of the 4-color theorem is a typical exercise for a decent computerized proof-checker.

Gonna find out who’s haughty or shite

Paul Erdös is coming to town

[spoiler]As many here know, Paul Erdös was a famous mathematician who specialized in graph theory, the subfield of math applicable to this problem.

When he encountered a particularly elegant proof, he’d say it was “straight from The Book.” The book in question is what Erdös imagined to be a collection of perfect mathematical proofs compiled by God.

Erdös died in 1996, so presumably he has his own copy of The Book now. Who better to evaluate a new proof?[/spoiler]

I have it on good authority that Erdös is presently conferring with Fermat and they are having a good, but marginal, laugh at our expense.

Hey, no fair of Fermat to steal a low number like that!

The wiki page cited by the OP mentions some conditions that must be met for the theorem to be true. One looked interesting:

If the map contains any regions of finite area but bounded by an infinitely long boundary (like a snowflake curve), all bets are off. It might require more than four colors.

Hm, I’d never realized that condition before. We know that for a two-dimensional map, four is always enough. And we know that for a three-dimensional map, there is no number of colors that’s enough: For any n, it’s easy to construct a three-dimensional map that requires n colors. Is there, then, some highest dimension for which four suffices? A dimension for which 5 is necessary and sufficient? A minimum dimension which requires an infinite number? I suspect that the answer to any of these is impossible to determine.

Why impossible to determine? If in 2 dimensions 4 colors suffice, and in 3 or more dimensions no finite number of colors suffice, that seems to cover all of your cases.

The Hadwiger-Nelson graph coloring problem remains unsolved. Perhaps OP is reacting to mention of this: A non-mathematician recently devised a Hadwiger-Nelson graph(*) that cannot be four-colored. It is the first such example known. (*=Two points in the plane are “adjacent” in Hadwiger’s notion if exactly one inch of distance separates them.)