# Four color map theorem...disproven? **URL:** \ **Category:** Factual Questions\ **Created:** [December 23, 2019, 9:02pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308 "2019-12-23T21:02:49Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [December 23, 2019, 9:02pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/1 "2019-12-23T21:02:49Z")
The four color map theorem: > **[Four color theorem](https://en.wikipedia.org/wiki/Four_color_theorem)** > > In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any 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. It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was inf Th... states that for a 2D area broken up into irregularly shaped borders, only four different colors need to shade in the borders. Would not this shape: [![](https://i.ytimg.com/vi/Y30VF3cSIYQ/maxresdefault.jpg) ](https://i.ytimg.com/vi/Y30VF3cSIYQ/maxresdefault.jpg) prove you need FIVE colors? ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 23, 2019, 9:07pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/2 "2019-12-23T21:07:10Z")
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. ---
**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\ **Post date:** [December 23, 2019, 9:10pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/3 "2019-12-23T21:10:16Z")
> [@Enola\_Straight](#): > > Would not this shape: > [https://i.ytimg.com/vi/Y30VF3cSIYQ/maxresdefault.jpg](https://i.ytimg.com/vi/Y30VF3cSIYQ/maxresdefault.jpg) > prove you need FIVE colors? Red and blue aren’t in contact with each other, so they don’t need to be different colors. ---
**Author:** ![Atamasama](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/atamasama/32/12961_2.png) [@Atamasama](https://boards.straightdope.com/u/Atamasama)\ **Post date:** [December 23, 2019, 9:10pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/4 "2019-12-23T21:10:20Z")
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. ---
**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\ **Post date:** [December 23, 2019, 9:26pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/5 "2019-12-23T21:26:09Z")
> [@Atamasama](#): > > 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. ---
**Author:** ![Kron](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kron/32/12444_2.png) [@Kron](https://boards.straightdope.com/u/Kron)\ **Post date:** [December 23, 2019, 9:37pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/6 "2019-12-23T21:37:37Z")
[Beautiful image made by me](https://imgur.com/Z5EeVDC) Two unconnected black regions, two unconnected blue regions, 4 each unconnected green and yellow regions. (background assumed black) ---
**Author:** ![Atamasama](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/atamasama/32/12961_2.png) [@Atamasama](https://boards.straightdope.com/u/Atamasama)\ **Post date:** [December 23, 2019, 9:58pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/7 "2019-12-23T21:58:15Z")
> [@Kron](#): > > [Beautiful image made by me](https://imgur.com/Z5EeVDC) > > Two unconnected black regions, two unconnected blue regions, 4 each unconnected green and yellow regions. > > (background assumed black) That’s well done. 🙂 ---
**Author:** ![Dewey\_Finn](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dewey_finn/32/4222_2.png) [@Dewey\_Finn](https://boards.straightdope.com/u/Dewey_Finn)\ **Post date:** [December 23, 2019, 10:06pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/8 "2019-12-23T22:06:03Z")
[Here](http://www.straightdope.com/columns/read/117/did-they-finally-solve-the-four-color-map-problem/) is a 1984 Cecil Adams column about the four-color map problem, talking about a 1976 solution. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [December 23, 2019, 11:31pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/9 "2019-12-23T23:31:41Z")
Hrumph. ☹ ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [December 24, 2019, 12:40am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/10 "2019-12-24T00:40:12Z")
> [@Enola\_Straight](#): > > Hrumph. > > ☹ Never challenge a math theorem without thinking it through twice, and then one more time. 😉 ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [December 24, 2019, 1:26am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/11 "2019-12-24T01:26:39Z")
> [@naita](#): > > 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. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 24, 2019, 1:33am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/12 "2019-12-24T01:33:14Z")
Indeed, verifying a proof of the 4-color theorem is a typical exercise for a decent computerized proof-checker. ---
**Author:** ![EdelweissPirate](https://avatars.discourse-cdn.com/v4/letter/e/8491ac/32.png) [@EdelweissPirate](https://boards.straightdope.com/u/EdelweissPirate)\ **Post date:** [December 24, 2019, 2:08am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/13 "2019-12-24T02:08:27Z")
> [@Senegoid](#): > > 'Tis the season for making a proof, checking it twice. 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] ---
**Author:** ![Musicat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/musicat/32/20189_2.png) [@Musicat](https://boards.straightdope.com/u/Musicat)\ **Post date:** [December 24, 2019, 2:55am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/14 "2019-12-24T02:55:11Z")
> [@EdelweissPirate](#): > > Gonna find out who’s haughty or shite > Paul Erdös is coming to town 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. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 24, 2019, 2:58am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/15 "2019-12-24T02:58:03Z")
Hey, no fair of Fermat to steal a low number like that! ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [December 24, 2019, 2:59am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/16 "2019-12-24T02:59:12Z")
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](https://en.wikipedia.org/wiki/Koch_snowflake)), all bets are off. It might require more than four colors. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 24, 2019, 4:17am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/17 "2019-12-24T04:17:27Z")
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. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 24, 2019, 4:28am UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/18 "2019-12-24T04:28:03Z")
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. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [December 24, 2019, 12:31pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/19 "2019-12-24T12:31:45Z")
> [@Chronos](#): > > 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. Note that a torus requires 7 colors [Torus Coloring -- from Wolfram MathWorld](http://mathworld.wolfram.com/TorusColoring.html) ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [December 24, 2019, 12:39pm UTC](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308/20 "2019-12-24T12:39:47Z")
[The Hadwiger-Nelson graph coloring problem](https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_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.) [Next page](https://boards.straightdope.com/t/four-color-map-theorem-disproven/845308.md?page=2)