[JadeNB]

> [1] Almost because the Kingdom of the Netherlands isn't a country.

Neither is the North Sea! Nonetheless, this is a neat example.

> Back to my original question: does anybody know of a valid counterexample for the statement on countries?

A standard counterexample to the hypotheses of the 4-colour theorem (though not to the conclusion, as consulting a map easily verifies) is Michigan, which is not connected.

The 4-colour theorem's hypotheses also rules out the possibility of 4 countries meeting at a corner (or, rather, declare that they don't meet in that case). If there were such an arrangement—and I'd be surprised if there isn't; for 3 countries, one has the example of Finland, Sweden, and Norway—then it would be easy to juice it up to a counterexample.

Maybe the guy who established an island with a bizarre currency, including one coin that had a denomination of π (I can't remember who—I thought Dean Kamen, but his Wikipedia page doesn't mention it), could be induced to subdivide his island in such a way as to create a counterexample. :-)

[Someone]

According to http://en.m.wikipedia.org/wiki/Quadripoint, there currently isn't any quadripoint between countries, although it is awfully close between Namibia, Botswana, Zambia and Zimbabwe.

Trijunctions are very common. Luxembourg has 3 of them, Germany (if I count correctly) 7.

[JadeNB]

Wow—regardless of its use to answer the question, that's a new word for me, and a fascinating article. (Who would have expected such a debate to lead to literal shots fired? Well, I guess any student of cartography and history, but not me.) Thanks!

[dedward]

I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours.

[JadeNB]

> I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours.

Corners matter (or do not matter, depending on how you like to phrase it!) only in the slightly non-intuitive sense that countries that meet only at a corner must be declared not to 'meet' at all for purposes of the theorem. Otherwise, you can have four regions meeting only at a corner (say by sub-dividing a square), which therefore use 4 colours, and a 'moat' surrounding them all, which therefore requires a 5th colour.

[Someone]

Or just n>4 regions meeting in a corner. Allowing 'meeting in a single point' turns the theorem into the 'infinite color theorem', which would be uninteresting.

[dedward]

Right.

The point I'm making is that this isn't simply that "there isn't anything in our current geo-political makeup where enough countries meet to disprove the theorem"... the theorem holds up under any theoretical design.

[JadeNB]

> The point I'm making is that this isn't simply that "there isn't anything in our current geo-political makeup where enough countries meet to disprove the theorem"... the theorem holds up under any theoretical design.

No, it doesn't! Of course, as a true theorem, its conclusion holds whenever its hypotheses do—but there are hypotheses, not all of which are satisfied by any theoretical (or, perhaps more importantly, real-world) design.

One is that meeting at a corner is not counted as 'meeting', which you may fairly object is just a definition rather than a hypothesis; but another, which is a genuine hypothesis rather than just a matter of definition, is that the countries / regions must be connected [0]. (For a real-world example where this hypothesis is not satisfied—although the conclusion still is—on the level of states, see Michigan; and, on the level of countries, see the US and Alaska. I don't know if it is possible to make all countries connected by building imaginary land bridges across non-territorial waters, but I doubt it.)

EDIT: As Someone pointed out at the top of this thread (http://news.ycombinator.com/item?id=8801082 and then, in response to my confusion, http://news.ycombinator.com/item?id=8804876), this disconnectedness can actually create an issue; the fact that it technically isn't an issue for the particular case that he or she mentions is just because the regions involved aren't technically 'countries', rather than because no pathological arrangement of countries can break the 4-colour theorem.

EDIT: [0] Probably I should say 'open and connected', to avoid the pathologies a malicious topologist could cook up.