April 12th, 2010, 04:30 PM  #1 
Senior Member Joined: Jan 2009 Posts: 344 Thanks: 3  number theory
The points in the plane are each colored blue, red, or yellow.Prove that there are two points of the same color of mutual distance unity. 
May 9th, 2010, 03:35 AM  #2 
Newbie Joined: Apr 2010 From: kenya Posts: 28 Thanks: 0  Re: number theory
I don't understand the meaning of mutual distance unity,but if at all I understand it then I can define my mutual distance for two points X,Y to be d(X,Y)=max{XiYi} for i=1,2.Now,consider the points (0,0),(0,1),(1,0),(1,1).Pick any three of these points and assume they are of different colors,the forth point has to be of the same color to any of the three.To generalize,for any number d consider the points (d,d),(d,d+1),(d+1,d),(d+1,d+1).

May 9th, 2010, 07:53 AM  #3 
Senior Member Joined: Apr 2008 Posts: 435 Thanks: 0  Re: number theory
Alternatively, consider a point P1. Say it's red. And consider the unit circle around it. All points on this circle must be either blue or yellow, and there will be both (I'm assuming, for the sake of contradiction, that no such points exist). Now, we are going to consider a few such circles. Choose a blue point P2 on this circle, which I now denote C1, and create C2. We note that where C1 and C2 intersect (at P3 and P4), there must be yellow points. Finally, and I'm sorry for the awkward configuration of circles, repeat to create C3 and C4 around P3 and P4. Good. Now, consider a point O on circle C4 such that the two points of intersection between the unit circle C5 around O and the unit circle C3 is unit length. We can do this because of the continuity of the implied function and the continuity of circles. Now, instead of finishing the problem (although it's now done), I consider instead a much easier question. Say we have colored the plane with a mere two colors, and we are to show the same thing. Well, choose an equilateral triangle in the plane. 3 points, 2 colors, and we are done immediately. Is this sufficient? 
May 10th, 2010, 07:43 AM  #4 
Newbie Joined: Apr 2010 From: kenya Posts: 28 Thanks: 0  Re: number theory
Smart.I didn't understand the last part when circle C5 came in until I took a pen and sketched your argument.I think it is sufficient.Wise thought.


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