Worked solution: The polynomial method resolves the cap set problem (Croot-Lev-Pach, Ellenberg-Gijswijt, 2016)

A cap set is a subset $A\subset\mathbb{F}_3^n$ containing no three distinct points $x,y,z\in A$ with $x+y+z=0$; since $-x=x+x$ holds only in characteristic $3$, this is exactly the condition that no three distinct elements form an affine line (equivalently, a three-term arithmetic progression $x,x+d,x+2d$ with $d\ne0$). Write $r_3(\mathbb{F}_3^n)=\max|A|$.

The trivial bound $|A|\le3^n$ was the only exponential statement available for decades: Meshulam (1995) proved $|A|=O(3^n/n)$ by Fourier analysis on $\mathbb{F}_3^n$, and Bateman–Katz (2012) pushed this to $O(3^n/n^{1+\epsilon})$ — still within a polynomial factor of trivial. On the other side, the largest known constructions (Edel, and improvements by Tyrell) gave only about $2.2^n$, so even the qualitative question 'is the growth rate strictly less than $3$?' was open.

Ellenberg–Gijswijt (2017), building on the new polynomial-method breakthrough of Croot–Lev–Pach (2016) for $(\mathbb{Z}/4\mathbb{Z})^n$, settle that question: $|A|\le O(c^n)$ with an explicit $c=3e^{-I(2/3)}\approx2.756$. The rest of this proof turns a cap set into a vanishing-polynomial problem and lets a dimension count do the work.