https://www.quantamagazine.org/20160531 ... maticians/
Simple Set Game Proof Stuns Mathematicians
this is the part that caught my eye
could this help with your projective geometry?
Game, Set, Match
To find an upper bound on the size of cap sets, mathematicians translate the game into geometry. For the traditional Set game, each card can be encoded as a point with four coordinates, where each coordinate can take one of three values (traditionally written as 0, 1 and 2). For instance, the card with two striped red ovals might correspond to the point (0, 2, 1, 0), where the 0 in the first spot tells us that the design is red, the 2 in the second spot tells us that the shape is oval, and so on. There are similar encodings for versions of Set with n attributes, in which the points have n coordinates instead of four.
The rules of the Set game translate neatly into the geometry of the resultingn-dimensional space: Every line in the space contains exactly three points, and three points form a set precisely when they lie on the same line. A cap set, therefore, is a collection of points that contains no complete lines.