# Is this triangular tiling problem decidable?

Hi there! The tiling problem below is relevant to my logic thesis. Do you have any idea on how to tackle it? Or can you refer me to people who might be able to help?

A triangle tiling is a finite set \mathcal{T}\subseteq\mathcal{P}(\mathbb{R}^2) of triangles such that

• \cup\mathcal{T} is a triangle;
• for any T,T'\in\mathcal{T} we either have
• T\cap T'=\emptyset,
• T\cap T'=\{v\} for some point v that is a corner vertex of both T and T',
• T\cap T' is a full edge of both T and T', or
• T=T'.

Equivalently, \mathcal{T} is a triangle tiling iff there exists a twodimensional simplicial complex \Sigma\subseteq\mathcal{P}(\mathbb{R}^2) such that the carrier of \Sigma is a triangle and \mathcal{T} is the set of triangles in \Sigma.

Suppose that G=(V,E) is a finite undirected graph and B\subseteq V.
We say that a triangle tiling \mathcal{T} is (G,B)-colored by a map f:\mathcal{T}\rightarrow V if

• for any T,T'\in\mathcal{T} such that T\cap T' is an edge of T and T', we have \big\{f(T),f(T')\big\}\in E;
• for any T\in\mathcal{T} we have f(T)\in B iff T intersects the boundary of \cup\mathcal{T};
• f is surjective.

We wonder whether the following problem is decidable: given a finite undirected graph G=(V,E) and B\subseteq V, is there a triangle tiling \mathcal{T} that can be (G,B)-colored?

The picture exemplifies this problem with an instance for which the answer is yes’'.

can’t you just alternate red and blue starting from a corner?

If I solve this for you, will I get awarded your degree instead of you receiving it?

Difficult to take academia seriously these days, in the future we are probably going to have a lot of people with degrees who had AI write it for them.

If you are an academic in 2023, I will assume you are somebody who fakes their way around.

Yeah, I think you’re right and there’s some sort of trivial solution. Can’t find an option in the interface to delete the thread…

No, I would still be the one to receive the degree but you would get an honorary mention if the problem is included in the thesis.

That’s not really a good motivation. Getting an honorary mention is not really worth the effort of studying a math problem someone else is going to use to specifically advance their own career. It’s not like this is going to help anyone except for your personal career.

I do not at all mean to discourage this particular question, only the added context of the thesis makes it less likely that I would want to think about it, and perhaps there are other people who have similar thoughts. Academia is just really ugly.

Good luck with finding the solution, I don’t think you need to worry about deleting it, that is an unnecessary worry/concern.