Let $r(n):=r_3(\mathbb{F}_3^n)=\max\{|A|: A \subset \mathbb{F}_3^n, \ A \text{ is 3-AP-free}\}$.
Edel proved that $r(n)\geq 2.217^n$ for sufficiently large $n$. His proof is by giving a construction of a cap-set $A$ in $\mathbb{F}_3^{480}$. Then observing that $A^k \subset \mathbb{F}_3^{480k}$ is also a cap set, that is,
$$r(480k)\geq |A^k|=|A|^k.$$
Is this the best known lower bound? Are there other known approaches to this problem other than construction in low dimension and then using this product argument?
Is this product argument expected to be the best we could expect? That is, do we hope to construct an $A$ such that this argument is tight?
I'd appreciate any references or answers to some of these questions.
Any help on the tags or on how to better ask these questions would be nice also.
Best Answer
I've just proved a new lower bound of $2.218^n$, in my paper 'New lower bounds for cap sets': https://arxiv.org/abs/2209.10045.
My new bound comes from extending Edel's ideas, with better computational methods (including a SAT solver) and introducing a new theoretical construction. I also conjecture that a lower bound of $2.233^n$ is possible, which is explained in section 4 of my paper.
My lower bound comes from a cap set in $\mathbb{F}_3^{56232}$, although I do also give cap sets in 396, 420 and 462 dimensions which beat Edel (slight correction to your post: Edel's bound comes from a cap set in $\mathbb{F}_3^{480}$).