Cap set $r(5)$

Cap set of 45 points, no three on a line, in 5-dimensional space over the field with 3 elements, drawn as nested 3 by 3 grids

The size of the largest set of points in $\mathbb{F}_3^{5}$ with no three on a line.

Value: $45$ 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: $45$ Checked construction
cap-set-5-lower-45, submitted by daniel-munro, 2026-11-05
Upper bound: None yet

Updates from the literature

1973 Lower bound: $45$
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 Caps, Yves Edel, 2010-04-28]
2002-07 Upper bound: $45$
Y. Edel, S. Ferret, I. Landjev and L. Storme, The classification of the largest caps in AG(5,3), Journal of Combinatorial Theory, Series A 99 (2002) 95-110.
[via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]