I'm trying to come up with a good Ramsey Theory question (details)

[jdinatale]

I have this really cool idea of asking Ramsey Theory questions on tilings (tessellations).

Classical Ramsey theory asks what is the minimum number of people one needs to invite to a party in order to have that every person knows m mutual friends or n mutual strangers.

I was thinking about coloring a tiling red and blue and asking similar questions, but I'm not sure what would be interesting. I have thought about this for a couple of weeks.

Basically, Ramsey Theory is simply asking "how many elements of some structure must there be to guarantee that a particular property will hold?"

I'm just not sure how to come up with an interesting question!

 

[Simon Bridge]

What - like "what is the minimum number of colors of tile needed so you can cover a surface so that no two adjacent tiles share the same color?" But something that requires students to exploit tesselation ideas?

OR: "How many trainee chefs in a kitchen before Ramsey is guaranteed to throw a wobbler?"
... ah... wrong Ramsey...

 

[Stephen Tashi]

The most elmentary questions are whether there is some way to represent graphs as tilings and vice-versa.

It seems clear than any colored tiling could be represented as a colored graph by using an edge connection between two nodes to represent two tiles sharing a common side. That representation would not capture the property of two tiles only sharing a common vertex. Is there a graph that also represents that property?

And given a colored graph, can it be represented as a tiling? I suppose that requires a rigorous definition of what a tiling is.

 

[jdinatale]

The question I had, which didn't work, is this: Given positive integers m and n, does there exist a tiling such that for ANY two-coloring of the tiling, every tile has m friends OR n strangers. (In this case two tiles are friends if they are adjacent and share a color and strangers if they are adjacent and do not share a color)

But it doesn't work because even starting with the extremely basic equilateral triangle tiling, and low numbers such as m = 2, and n = 3, we always find counterexamples.

What - like "what is the minimum number of colors of tile needed so you can cover a surface so that no two adjacent tiles share the same color?" But something that requires students to exploit tesselation ideas?

OR: "How many trainee chefs in a kitchen before Ramsey is guaranteed to throw a wobbler?"
... ah... wrong Ramsey...

I'm pretty sure the answer to the first question is four via the four color map theorem, no?

The most elmentary questions are whether there is some way to represent graphs as tilings and vice-versa.

It seems clear than any colored tiling could be represented as a colored graph by using an edge connection between two nodes to represent two tiles sharing a common side. That representation would not capture the property of two tiles only sharing a common vertex. Is there a graph that also represents that property?

And given a colored graph, can it be represented as a tiling? I suppose that requires a rigorous definition of what a tiling is.

The answer to the second question I'm guessing is no because what if you had a colored, say K_6. I don't think it would be possible to have all 6 tiles adjacent to every other tile at once. I will think about it.

 

[CellCoree]

One possible Ramsey Theory question related to tilings could be: What is the minimum number of tiles needed to guarantee that every possible tiling of a given shape contains at least one tile of each color (red and blue)? This question would be similar to the classic Ramsey problem, but applied to tilings instead of people and friendships. It could also be extended to consider different shapes and colors, adding more complexity to the problem. Additionally, exploring the relationship between the number of tiles and the number of colors needed to guarantee the property could also be an interesting avenue for further investigation.