In this post I give a question for you to think about.
My next post will have the answer and the proof.
1) The following are known and I have a set of slides about it here
a) For all 2-colorings of \(R^2\) there exists two points an inch apart that are the same color. (You can do this one.)
b) For all 3-colorings of \(R^2\) there exists two points an inch apart that are the same color. (You can do this one.)
c) For all 4-colorings of \(R^2\) there exists two points an inch apart that are the same color. (You cannot do this one.)
2) SO, lets look at other shapes
A unit square is square with all sides of length 1.
Given a coloring of \(R^d\) a mono unit square is a unit square with all four corners the same color.
a) There is a 2-coloring of \(R^2\) with no mono unit square. (You can do this one.)
b) What is the value of d such that
-- There is a 2-coloring of \(R^d\) with no mono unit square.
-- For all 2-colorings of \(R^{d+1}\) there is a mono unit square.
My next post will tell you what is known about this problem.
Until then, you are invited to think about it and see what you can find out. Perhaps you will get a better result then what is known since you are untainted by conventional thinking. Perhaps not.
Feel free to leave comments. However, if you don't want any hints then do not read the comments.
Here is a simple argument that in R^6 there is a mono unit square. First, take a unit simplex. By the pigeonhole principle, this will have 3 vertices of the same color, say, red - denote them by T and assume that one of them is the origin o. Take a unit simplex S on 5 vertices in the 4-dim subspace perpendicular to the plane of T such that one of its vertices is o. Consider the coloring of S x T. This consists of 5 copies of T, one of them being the original T. If in any of the other 4 copies we have two red vertices, then these form a mono square with T. Otherwise, in each copy we have two blue vertices. In two copies, these two vertices will be the 'same' vertices of T, so they form a mono squre.
ReplyDeleteBravo! I will be presenting a different proof, and yours, when I post on Wedenday.
ReplyDeleteDom- email me privately- I want to email you the proof I will present to see if its the same are your proof or different. I think its different but want to ask you. gasarch@umd.edu
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