Green's Open Problems #17

### What is the conjecture
https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#problem.17

Suppose that $A \subset F^n_3$ is a set of density $\alpha$. Under what conditions on $\alpha$ is $A$ guaranteed to contain a 3-term progression with nonzero common difference in $\\{0,1\\}^n$? For fixed $\alpha$, must $A$ contain $≫_α 6^n$ such progressions?

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-05
* ams-11

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Start by reading Green's Open Problems PDF at the linked Problem 17, then inspect existing conjecture formalizations in this repository for the expected structure. Done means the stated conjecture has been added as a Lean formalization and fits the repository's conventions.

Written by the indexing model from the issue text.