Cap set problem

Determine whether the maximum size $r_3(\mathbb{F}_3^n)$ of a cap set—a subset $A \subseteq \mathbb{F}_3^n$ containing no three-term arithmetic progression (no three distinct elements $x, y, z \in A$ with $x + y + z = 0$)—grows exponentially with a base strictly less than $3$, i.e., $r_3(\mathbb{F}_3^n) \le c^n$ for some constant $c < 3$.

In May 2016, Ernie Croot, Vsevolod Lev, and Péter Pál Pach introduced a breakthrough polynomial method to bound progression-free sets in $(\mathbb{Z}/4\mathbb{Z})^n$. Within days, Jordan Ellenberg and Dion Gijswijt adapted their technique to $\mathbb{F}_q^n$, proving that every cap set in $\mathbb{F}_3^n$ has size at most $O(c^n)$ with $c = 3(8^{-1/3})(1+8^{-1/3}+8^{-2/3}) / 2 \approx 2.7552 < 3$ (equivalently $\frac{3}{8^{1/3}} \cdot \dots \approx 2.756$). Terence Tao reformulated the argument cleanly via the slice rank of tensors. While this settled the qualitative exponential-saving conjecture, pinning down the exact exponential base between the best lower bound ($\approx 2.220^n$) and $2.7552^n$ remains open.