Undergraduate Math Club Seminar

In the card game "SET", we can choose up to 20 cards that contain no set among them. The cap set problem asks for the same bound for a generalized version of SET: if we collect all cards that have n distinct fixed attributes, each of which taking 3 possible values, what is the largest number of cards we can choose that does not contain a set? In 1995, Meshulam proved that this number is O(1 / n); in 2012, Bateman and Katz lowered this bound to O(1 / n ^(1 + ε)) for some ε > 0. Then in 2016, the bound was dramatically improved to O(e ^(-kn)) for some constant k. In this talk, we will see how to prove this exponential bound.