In my last post (see here) I invited you to work on the following question:
Find a \(d\) 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.
Actually I should have phrased my question as What do we know about d?
Here is what we know
a) \(d \ge 2\). There is a 2-coloring of \(R^2\) with no mono unit square. This is easy and I leave to you.
b) \(d\le 5\). For all 2-colorings of \(R^6\) there is a mono unit square. I will give pointers to the relevant papers and to my slides later in this post.
c) \(d\le 4\). For all 2-colorings of \(R^5\) there is a mono unit square. This is by an observation about the proof for \(R^6\). It will be in the slides about \(R^6\).
d) \(d\le 3\). This is in a paper that the reader Dom emailed me a pointer to. Dom is better at Google Search than I am. The link is here.
MY SLIDES:
\(K_6\) is the complete graph on 6 vertices. We will be looking at 2-colorings of its edges
\(C_4\) is the cycle on 4 vertices. A mono \(C_4\) has all four edges the same color.
We need a result by Chvtal and Harary in this paper here.
Lemma: For all 2-colorings of the edges of \(K_6\) there is a mono \(C_4\).
The proof appears both in their paper, here, and on slides I wrote here.
Stefan Burr used this to prove the following theorem.
Thm: For all 2-colorings of \(R^6\) there is a mono unit square.
The proof was appears (with credit given to Stefan Burr) in a paper by Erdos, Graham, Montgomery, Rothchild, Spencer, Straus, here, and on slides I wrote here.
Random Points
1) It is open what happens in \(R^3\).
2) The proof for \(R^6\) uses very little geometry. Dom had a proof for \(R^6\) in a comment on my last post that used geometry. The proof for \(R^4\) uses geometry.
3) An ill-defined open question: Find a proof that every 2-coloring of \(R^4\) has a mono unit square that does not use that much geometry and so I can make slides about it more easily.
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