The game called "Set", in case you don't know, is a deck of 81 cards, each one having a visual image on it. The image displays 4 characteristics, each of which is a choice of one of 3 possible versions. The characteristics happen to be: Color, Number, Shape, and Shading. Color = {Red, Green, Purple}; Number = {One, Two, Three}; Shape = {Oval, Diamond, Squiggle}; Shading = {Light, Medium, Heavy}. All of which details are irrelevant to the math behind it. We may as well assume the 81 cards is the vector space V = (F_3)^4 of dimension 4 over the field of 3 elements. The goal of the game is to find affine lines in V, namely subsets of the form L + v where L is any 1-dimensional subspace of V and v is any element of v. These are all, of course, subsets of size 3. DIFFICULT PROBLEM: What is the largest size of a subset of V that contains no affine line? The answer is 20. Every known proof of this is a tedious case-by-case analysis or else by computer search. GENERALIZED DIFFICULT PROBLEM: Let V_n denote the vector space (F_3)^n. What is the largest size of a subset of V_n containing no affine line? Let M(n) denote this number: the maximum size of any subset of V_n containing no affine line. Apparently even the n=5 case borders on being too large for computers to handle in any reasonable time, but that has been solved. In fact, here is what is known: n 1 2 3 4 5 6 7+ M(n) 2 4 9 20 45 112-114 ? ---------------------------------------------------------- QUESTION: Is there a nice asymptotic expression for M(n) ? --Dan