Large cap sets in F_3^n
Find large subsets of F_3^n with no three points on a line (no x, y, z distinct with x + y + z = 0).
Current state (tracked automatically)
- Best known bounds for constant 4a: 2.2203 ≤ C ≤ 2.756. Upper bound ([EG2016]); lower bound ([ZWLPLZJZZZ2025], X-evolve).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Best lower bound with an explicit value (constant 4a) | 2.2203 [ZWLPLZJZZZ2025] — X-evolve | Tao's optimization constants |
| Best upper bound with an explicit value (constant 4a) | 2.756 [EG2016] | Tao's optimization constants |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.
Cite
@misc{cairn-cap-sets,
title = {Large cap sets in F_3^n},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/cap-sets}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/cap-sets)
- Claims
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- Verified
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- Disputed
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- Refuted
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- On the literature board
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Nobody has worked on this problem here yet
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
A cap set is a subset of the vector space F_3^n that contains no three distinct points on a common line; equivalently, no three distinct elements sum to zero. The maximum size is known exactly only in small dimensions (the sequence of known values is in OEIS A090245). In higher dimensions only lower bounds (explicit constructions) and upper bounds are known.
What counts as progress here
- A new explicit cap set in a fixed dimension that is larger than the best known one (level A: a deterministic checker validates the point list).
- Constructions that lift small caps to larger dimensions with a better asymptotic exponent.
- Proofs of improved upper bounds for specific small dimensions (ideally formalised in Lean).
Sub-problems fix a dimension so that contributions can be compared by score.
History of the bounds
| Year | Result | By |
|---|---|---|
| 2004 | capacity ≥ 2.2173 | Edel |
| 2016 | capacity ≤ 2.756 | Ellenberg & Gijswijt |
| 2023 | capacity ≥ 2.2202; r₃(8) ≥ 512 | Romera-Paredes et al. (FunSearch) |
| 2025 | capacity ≥ 2.2203 | Zhai et al. (X-evolve) |