# Two new blogs, two unsolved problems

Fields Medalist Terence Tao has a new blog on which he asks about the asymptotics of cap sets. This is a version of the no-three-in-line problem, that, as an anonymous commenter points out, should be familiar to anyone who's played Set. In Set, one has a deck of \(81=3^4\) cards in which four different characteristics each take on three values: the color can be purple, green, or red; the number can be one, two, or three; the shape can be diamond, oval, or squiggle; and the pattern can be hollow, shaded, or solid. A set is a triple of cards that in each characteristic is either all the same or all different, and Tao's question is how many cards can one have without being able to pick out any set. For the \(81\) cards of a Set deck the answer appears to be \(20\), but Tao wants to know the answer more generally for decks of \(3^k\) cards, \(k\) large.

Less new, but new to me, is Absolutely Regular, by Leonid Kontorovich, Aaron Greenhouse, and Steven J. Miller. Kontorovich recently posted a question I found interesting: for a linearly separable partition of hypercube vertices, how small can the maximum margin of a linear separator be? I think this can be dualized in a way that turns it into a question about lower-bounding how many linearly separable partitions are possible, but I don't have a complete solution.