tag:blogger.com,1999:blog-9124539381685751273.post2530131314303950556..comments2023-06-19T04:35:06.263-07:00Comments on Skeptic's Play: A painted plane IIImillerhttp://www.blogger.com/profile/05990852054891771988noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-9124539381685751273.post-3490569936027734602008-10-16T10:58:00.000-07:002008-10-16T10:58:00.000-07:00This problem (and the other painted plane problems...This problem (and the other painted plane problems) are questions that relate to the branch of mathematics called Ramsey Theory. Ramsey Theory has a reputation for having complicated proofs involving insanely large numbers (eg <A HREF="http://en.wikipedia.org/wiki/Graham%27s_number" REL="nofollow">Graham's number</A>). I do not know whether current Ramsey Theory can answer your questions.<BR/><BR/>The rectangle problem, though, requires much smaller numbers. (Hint!) 21 points will suffice to show that a single-colored rectangle must exist on a 2-colored plane.millerhttps://www.blogger.com/profile/05990852054891771988noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-2913269659060355582008-10-16T09:08:00.000-07:002008-10-16T09:08:00.000-07:00Since you're posing the problem, I'm going to assu...Since you're posing the problem, I'm going to assume (since I lack the time or mathematical ability to actually <I>solve</I> the problem) that the solution is not just a trivial extension of the triangle problem.<BR/><BR/>Which then raises the question: is there a general solution for the m/n (finite) problem, where you have a plane painted with m colors, and you have to prove you can draw a regular n-gon with all corners (vertices?) of the same color.<BR/><BR/>How about any definable shape/curve with a finite number of control points?<BR/><BR/>How about fractal shapes and curves with infinite control points that are non-space-filling?<BR/><BR/>(I suspect it's trivial to show that a space-filling fractal cannot have all of its control points of one color.)Larry Hamelinhttps://www.blogger.com/profile/08788697573946266404noreply@blogger.com