Cap set

The 9 points and 12 lines of , and a 4-element cap set (the four yellow points) in this space

In affine geometry, a cap set is a subset of (an -dimensional affine space over a three-element field) where no three elements sum to the zero vector. The cap set problem is the problem of finding the size of the largest possible cap set, as a function of .[1] The first few cap set sizes are 1, 2, 4, 9, 20, 45, 112, ... (sequence A090245 in the OEIS).

Cap sets may be defined more generally as subsets of finite affine or projective spaces with no three in line, where these objects are simply called caps.[2] The "cap set" terminology should be distinguished from other unrelated mathematical objects with the same name, and in particular from sets with the compact absorption property in function spaces[3] as well as from compact convex co-convex subsets of a convex set.[4]

  1. ^ Austin, David (August 2016), "Game. SET. Polynomial.", Feature column, American Mathematical Society.
  2. ^ Cite error: The named reference edel was invoked but never defined (see the help page).
  3. ^ See, e.g., Chapman, T. A. (1971), "Dense sigma-compact subsets of infinite-dimensional manifolds", Transactions of the American Mathematical Society, 154: 399–426, doi:10.1090/s0002-9947-1971-0283828-7, MR 0283828.
  4. ^ See, e.g., Minʹkova, R. M. (1979), "Weak Korovkin spaces", Akademiya Nauk Soyuza SSR, 25 (3): 435–443, 477, MR 0534099.