Cap sets

A cap set is a subset of vectors over the three-element field containing no three distinct vectors in an arithmetic progression. Researchers seek large constructions and upper bounds as dimension increases.

Work in the vector space F₃ⁿ, where coordinates are 0, 1, or 2 and arithmetic is modulo 3. A cap set contains no three distinct vectors x, y, z with x + y + z = 0. This gives a concrete finite construction problem: include as many vectors as possible without introducing a forbidden triple.

A construction in one dimension supplies a lower bound on the maximum size there. A general construction can yield an asymptotic lower bound. An upper bound restricts every cap set. These are complementary kinds of progress, and finding a larger example does not establish optimality.