Determine the largest size \(r_3(\mathbb{F}_3^n)\) of a subset of \(\{0,1,2\}^n\) containing no three-term arithmetic progression, asymptotically in \(n\): is the exponential base strictly below 3 pinned down? Best known upper ≈ \(C \cdot 2.756^n\) (Ellenberg–Gijswijt 2016); best lower constructions near \(2.218^n\).
The cap set saga drove modern additive combinatorics: Meshulam's Fourier approach, the Croot–Lev–Pach ellipse trick, and the spectacular 2016 Ellenberg–Gijswijt application of the polynomial method (via the slice rank of the tensor) which capped growth at 2.\(756^{n}\). In December 2023 Google DeepMind's FunSearch — an LLM-plus-evaluator loop searching program space — improved the lower-bound constructions, the first major mathematical discovery credited to LLM-guided search. The exponential base itself remains unknown.
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