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Level A · Machine-checkable Combinatorics P-cap-sets-dim-8

Cap sets in dimension 8

Find a cap set in F_3^8 larger than the best known construction.

Status (as of 2026-10-03): Open. The best known cap in F₃⁸ has 512 points (FunSearch, Romera-Paredes et al., Nature 2023/24); no larger construction has been reported since.
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Cite 
@misc{cairn-cap-sets-dim-8,
  title        = {Cap sets in dimension 8},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/cap-sets-dim-8}},
  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

Hard Beats the record: A cap set in F₃⁸ (no three points on a line) with more than 512 points.

512 was found by FunSearch (2023/24) with program search over priority functions. Products and extensions of good caps in lower dimensions are the usual starting point.

  1. Get a search program — your chatbot writes it from the prompt below
  2. Run it on your computer — it checks its own result and writes a file
  3. 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.

Get a search program in ChatGPT ↗ Upload a construction

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

Or give your chatbot a small task on this problem instead
or , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

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

The problem

Fix n = 8. Program-search methods have recently improved the best known construction in this dimension (see the Nature paper below), which suggests further improvements may be reachable.

Submission format: as for dimension 7, with points given as strings of 8 digits from {0,1,2}. Score = number of points. Include the program that generated the construction as a code artifact so the result is also reproducible (level B), not only checkable.

History of the bounds

YearResultBy
2023r₃(8) ≥ 512Romera-Paredes et al. (FunSearch)

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