Cap set $r(6)$

Cap set of 112 points, no three on a line, in 6-dimensional space over the field with 3 elements, drawn as nested 3 by 3 grids

The size of the largest set of points in $\mathbb{F}_3^{6}$ with no three on a line.

Value: $112$ 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: $112$ Checked construction
cap-set-6-lower-112, submitted by daniel-munro, 2026-11-05
Upper bound: None yet

Updates from the literature

1973 Lower bound: $112$
R. Hill, On the largest size of cap in S_{5,3}, Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 54 (1973) 378-384.
[via B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M.P. Kumar, E. Dupont, F.J.R. Ruiz, J.S. Ellenberg, P. Wang, O. Fawzi, P. Kohli and A. Fawzi, Mathematical discoveries from program search with large language models, Nature 625 (2024) 468-475.]
2002-02-07 Upper bound: $114$
J. Bierbrauer and Y. Edel, Bounds on affine caps, Journal of Combinatorial Designs 10 (2002) 111-115.
[via Caps, Yves Edel, 2010-04-28]
2008-01-10 Upper bound: $112$
A. Potechin, Maximal caps in AG(6,3), Designs, Codes and Cryptography 46 (2008) 243-259.
[via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]