Here is a neat question that I stumbled upon during my academic random walk.
A sequence of vectors is called an arithmetic progression of length k if there exists a vector such that . In other words, an arithmetic progression in is exactly the generalization you would expect of your good ol’ high-school arithmetic progression in . For example, in , the sequence is an arithmetic progression of length 3, with difference vector .
An infinite path in is a sequence of distinct points such that . In other words, you start at some , and then progress by going one step up or down or left or right or forward or backward or whatever it is they call it in general dimensions, while making sure not to intersect yourself. For example, here is the beginning of an infinite path in , which also just happens to contain as a subset the arithmetic progression given above (shown in red):
Question: Does every infinite path in contain arbitrarily long arithmetic progressions? That is, given a path , is it true that it contains a subsequence which is an arithmetic progression, for every ?
I really liked this question, since at first I had no clue as to what the answer was; while working on it I kept changing my mind, at one time certain that arithmetic progressions are unavoidable, and at the next imagining that an example of a progressionless path is nearly within my grasp.
You too, O’ faithful invisible readers, can tell me what you think, in the form of a non-binding referendum! Post your thoughts and ideas in the comments; I will post the answer in the not-too-distant future.