Level A · Machine-checkable Combinatorics P-cap-sets
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).
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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 2026-09-28}
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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.