5/24/2019, Robert Won, University of Washington The card game SET, finite affine geometry, and combinatorial number theory

Robert Won, University of Washington
The card game SET, finite affine geometry, and combinatorial number theory  
The game SET is a card game of pattern-recognition. To play the game, twelve cards are dealt face up and all players look for SETs, which are collections of three cards satisfying a certain property. When a SET is found, it is removed and three new cards are dealt. The player who finds the most SETs is the winner. When playing the game, a natural question arises: does every collection of twelve cards contain at least one SET? Or, perhaps more precisely: how many cards are needed to guarantee the presence of a SET?   
This question is related to a problem that Terence Tao, in a blogpost from 2007, described as "perhaps [his] favourite open question." In this talk, we explore the connections between SET, finite affine geometry, and combinatorial number theory. We discuss recent breakthrough work of Ellenberg and Gijswijt which answers Tao's question. Finally, we introduce a generalization of this question and present some recent results.