Abstractions of the Game “SET”

Author

Martin Grant, San Jose State UniversityFollow

Publication Date

Spring 2025

Degree Type

Master's Project

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

Wasin So

Second Advisor

Edgar Bering IV

Third Advisor

Kyle Hambrook

Keywords

SET (game), Cap sets, Finite fields, Polynomial methods

Abstract

Each card in the game of SET can be represented as a point in Z⁴₃, where Z₃ is the f ield of 3 elements; an in-game position without any SETs can be represented as a cap set. We find the largest cap sets in Zⁿ ₃ for n ≤ 4 and prove their uniqueness. Then, we provide more insight into Ellenberg and Gijswijt’s proof of upper bound for the maximum size of cap set in Fⁿ_q, where F_q is the field of q elements, which they find to be o(cⁿ) for some c < q. Finally, we present evidence suggesting Ellenberg and Gijswijt’s method fails to prove c ≤ 0.841434q.

Recommended Citation

Grant, Martin, "Abstractions of the Game “SET”" (2025). Master's Projects. 1587.
https://scholarworks.sjsu.edu/etd_projects/1587