A cap set is a subset 
of the affine space 
containing no three distinct collinear points, where 
is a positive integer
and 
is the finite
field with three elements. Equivalently, the only
solutions in 
to
 |
have 
. This is also the condition that
contain no nonconstant three-term arithmetic progression, since 
is the same equation over
(Davis and Maclagan 2003, Ellenberg
and Gijswijt 2017).
The set 
is a cap set of size 
. Indeed, three entries from 
can have sum zero modulo 3 only
when they are all equal, so the vector equation forces
equality in every coordinate.
Let 
be the maximum
size of a cap set in 
.
Ellenberg and Gijswijt (2017) proved that there is a constant 
, independent of 
, such that
 |
Since 
, this is exponentially smaller
than the 
points of
the full affine space.
In dimension four, 
(Davis and Maclagan 2003). Under the SET
game correspondence, this means that 20 cards can contain no valid triple, whereas
every collection of 21 cards contains one.
with No Three-Term Arithmetic Progression." Ann.
Math. 185, 339-343, 2017.