For \(R^3\) the problem is open. That's too bad. We live in \(R^3\)

(If you live in Montgomery County Maryland OR if you care about Education, you MUST read this guest blog by Daniel Gottesman on Scott Aaronson's blog HERE.) 

(This post is a sequel to a prior post on this topic that was here. However, this post is self-contained---you don't need to have read the prior post.)  

(Later in the post I point to my open problems column that does what is in this post rigorously. However , that link might be hard to find, so here it is:  HERE)

BILL: I have a nice problem to tell you about. First, the setup.

Say you have a finite coloring of \(R^n\).

A  mono unit square is a set of four points that are

(a) all the same color, and

(b) form a square of side 1. The square does not need to be parallel to any of the axes.

DARLING: Okay. What is the problem?

BILL:  It is known that for all  2-colorings of \(R^6\) there is a mono unit square.

DARLING: \(R^6\)? Really! That's hilarious! Surely, better is known.

BILL: Yes better is known. And stop calling me Shirley.

DARLING: Okay, so what else is known?

BILL: An observation about the \(R^6\) result gives us the result for \(R^5\). (The \(R^5\) result also follows from a different technique.) Then a much harder proof gives us the result for \(R^4\). It is easy to  construct  a coloring of \(R^2\) without a mono unit square. The problem for \(R^3\) is open.

DARLING: That's too bad. We live in \(R^3\).

DARLING: Someone should write an article about all this including proofs of all the known results, open problems,  and maybe a few new things.

BILL: By someone you mean Auguste Gezalyan (Got his  PhD in CS, topic Comp Geom, at  UMCP), Ryan Parker (ugrad working on Comp Geom at UMCP), and Bill Gasarch (that's me!)  Good idea!

A FEW WEEKS LATER

BILL: Done! See here. And I call the problem about \(R^3\) The Darling Problem.

DARLING: Great! Now that you have an in-depth knowledge of the problem---

BILL: Auguste and Ryan have an in depth knowledge. Frankly I'm out of my depth.

DARLING: Okay, then I'll ask them:  What do you think happens in \(R^3\) and when do think it will be proved?

AUGUSTE: I think there is a 2-coloring of \(R^3\) with no mono unit square.

RYAN: I think that for every 2-coloring of \(R^3\) there is a mono unit square.

BILL: I have no conjecture; however, I think this is the kind of problem that really could be solved. It has not been worked on that much and it might just be one key idea from being solved. It is my hope that this article and blog post inspires someone to work on it and solve it. 

OBLIGATORY AI COMMENT

Auguste asked ChatGPT (or some AI) about the problem. It replied that the problem is open and is known as The Darling's Problem. This is rather surprising---Auguste asked the AI about this before I had submitted the article (it has since appeared) and before this blog post. So how did AI know about it? It was on my website.  I conjecture that Auguste used some of the same language we used in the paper so the AI found our paper. The oddest thing about this is that I don't find this odd anymore. 

 COLOR COMMENTARY  

The article appeared as a SIGACT News Open Problems Column. Are you better off reading it there or on my website, which is pointed to above. The SIGACT News version is (a) behind a paywall, and (b) in black and white. The version on my website is (a) free access, and (b) uses color. You decide.