Maximum number of sets with four properties for n > 12

Determine, with rigorous proofs, the exact maximum number of 3-card SETs achievable on a board of n cards drawn from the standard four-property SET deck (equivalently, among n points in the affine space F_3^4 where sets correspond to lines) for every integer n > 12.

Background

The paper proves exact maxima for the number of 3-card sets when using the standard four-property SET deck for all n from 3 to 12, culminating in the main result that the maximum for n = 12 is 14. The geometric framework models cards as points in F_3^4 and sets as lines, with magic squares (2-flats) playing a central role in the combinatorial structure.

Beyond n = 12, the authors do not have complete proofs for the four-property case. Although they provide computational methods, including exhaustive search for small n and a consecutive maximization heuristic, these do not settle the exact maxima for n > 12. Consequently, identifying and proving the exact maximum number of sets for each n > 12 remains unresolved.