The cap set constant

Pin down how fast the largest sets in F_3^n with no three points on a line can grow: between 2.2203^n and 2.756^n.

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1	# The cap set constant
2
3	How large can a set of points in $\mathbb{F}_3^n$ be if no three of them lie on a line? The largest such sets grow like $C^n$ for a constant $C$ known only to lie between 2.2203 and 2.756, and every improvement of the lower bound comes with an explicit object that a short program can check.
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5	## Problem
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7	A cap set is a set $A \subseteq \mathbb{F}_3^n$ that contains no line: there are no three distinct points $x, y, z \in A$ with $x + y + z = 0$. Equivalently, $A$ contains no three-term arithmetic progression. Let $r_3(n)$ be the largest size of a cap set in $\mathbb{F}_3^n$.
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9	The product of two cap sets is again a cap set, so $r_3(m+n) \ge r_3(m)\,r_3(n)$ and the limit
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11	$$C = \lim_{n \to \infty} r_3(n)^{1/n} = \sup_{n \ge 1} r_3(n)^{1/n}$$
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13	exists. Determine $C$, or narrow the known range $2.2203 \le C \le 2.756$.
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15	## Background
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17	The cap set problem is the finite-field version of a classical question about integers: how dense can a set be without three terms in arithmetic progression? Before 2016 the best upper bound, due to Bateman and Katz, was $O(3^n/n^{1+\epsilon})$, which does not improve the base 3. Ellenberg and Gijswijt then adapted a polynomial method of Croot, Lev and Pach to prove an exponential improvement in a five-page paper, and the constant is also tied to the sunflower problem (Naslund and Sawin). Lower bounds come from explicit constructions that glue small caps together; since 2023 several of them have been found with program search.
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19	## What is known
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21	- Upper bound $C \le 2.756$ (Ellenberg and Gijswijt, 2017; the trivial bound is 3).
22	- Lower bounds, in order: $2.1146$ (Pellegrino, 1970); $2.2101$ (Calderbank and Fishburn, 1994); $2.2173$ (Edel, 2004); $2.218$ (Tyrrell, 2023); $2.2202$ (Romera-Paredes et al., 2023, from a partial admissible set of size 237,984 found by FunSearch); $2.2203$ (Zhai et al., 2025, from a larger partial admissible set).
23	- Small dimensions. The exact value of $r_3(n)$ is known only for $n \le 6$. In $\mathbb{F}_3^8$, FunSearch found a cap set of 512 points, larger than any previously known.
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25	## What would count as progress
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27	- A cap set in a fixed dimension larger than the best known, for example more than 512 points in $\mathbb{F}_3^8$. Checking it takes well under a second: for each pair $x, y \in A$, test whether $-x-y$ is also in $A$.
28	- A construction certifying $C > 2.2203$: the small caps or admissible sets it uses, and the program that assembles them and computes the resulting exponent, so that both the object and the arithmetic can be rerun.
29	- An upper bound below $2.756$, given as a proof. This would need a genuinely new idea and is the harder direction.
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31	## References
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33	1. J. S. Ellenberg and D. Gijswijt, "On large subsets of $\mathbb{F}_q^n$ with no three-term arithmetic progression", Annals of Mathematics 185 (2017), 339–343. [doi:10.4007/annals.2017.185.1.8](https://doi.org/10.4007/annals.2017.185.1.8), [arXiv:1605.09223](https://arxiv.org/abs/1605.09223)
34	2. F. Tyrrell, "New lower bounds for cap sets", Discrete Analysis 2023:20. [Journal page](https://discreteanalysisjournal.com/article/91076-new-lower-bounds-for-cap-sets), [arXiv:2209.10045](https://arxiv.org/abs/2209.10045)
35	3. B. Romera-Paredes et al., "Mathematical discoveries from program search with large language models", Nature 625 (2024), 468–475, published online December 2023. [doi:10.1038/s41586-023-06924-6](https://doi.org/10.1038/s41586-023-06924-6)
36	4. Y. Zhai et al., "X-evolve: Solution space evolution powered by large language models" (2025). [arXiv:2508.07932](https://arxiv.org/abs/2508.07932)
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38	## Source
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40	Imported from [Optimization Constants in Mathematics, constant 4a](https://teorth.github.io/optimizationproblems/constants/4a.html), curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.