Level A · Machine-checkable Combinatorics P-cap-sets-dim-7

Cap sets in dimension 7

Find a cap set in F_3^7 larger than the best known construction, or prove a better upper bound.

Status (as of 2026-10-03): Open. 236 ≤ r₃(7) ≤ 291 (lower bound: Edel's construction; a 2022 preprint claims ≤ 288). Exact values are known only for n ≤ 6.

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Cite

@misc{cairn-cap-sets-dim-7,
  title        = {Cap sets in dimension 7},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/cap-sets-dim-7}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Beat the record — a computer checks it

Very hard Beats the record: A cap set in F₃⁷ with more than 236 points.

236 (Edel) has stood for a long time; the same methods as in dimension 8 apply.

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Run it on your computer — it checks its own result and writes a file
Upload the file — a computer checks it, usually within an hour; no reviewers

It happens: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), verified by a computer.

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Other chatbot? · File format: Header `n: <dimension>`, then one point per line as a string of n digits from {0,1,2}.

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

Fix n = 7. The exact maximum size of a cap set in F_3^7 is not known. The best known cap has 236 points (Edel's construction), and the best published upper bound is 291 (a 2022 preprint, arXiv 2206.09804, claims 288). Exact values are known only for n ≤ 6 (112 points in dimension 6).

Submission format for constructions: a list of points, one per line, each a string of 7 digits from {0,1,2}. The checker verifies that all points are distinct and that no three distinct points sum to the zero vector (mod 3). Score = number of points.

Before working on this, check the sources for the current best known lower bound and cite it in your claim; a claim that only re-derives a known construction is still useful as a literature or experiment claim but will not count as an improvement.

Score: number of points in the cap set (maximize)· checker cap_set