Cap set $r(3)$

Cap set of 9 points, no three on a line, in 3-dimensional space over the field with 3 elements, drawn as three 3 by 3 grids side by side

The size of the largest set of points in $\mathbb{F}_3^{3}$ with no three on a line.

Value: $9$ Verified lower bound

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

Lower bound: $9$ Checked construction
cap-set-3-lower-9, submitted by daniel-munro, 2026-11-05
Upper bound: None yet

Updates from the literature

1947 Lower bound: $9$
R.C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā 8 (1947) 107-166.
[via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]
1947 Upper bound: $9$
R.C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā 8 (1947) 107-166.
[via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]