Earlier this month, in the previous incarnation of this page, I posed a question which I thought was unsolved, and obtained the answer (in fact, it was solved 25 years ago) within a week. Now that this new version of the page has better feedback capability, I am now tempted to try again, since I have a large number of such questions which I would like to publicise. (Actually, I even have a secret web page full of these somewhere near my home page, though it will take a non-trivial amount of effort to find it!)
Perhaps my favourite open question is the problem on the maximal size of a cap set – a subset of ( being the finite field of three elements) which contains no lines, or equivalently no non-trivial arithmetic progressions of length three. As an upper bound, one can easily modify the proof of Roth’s theorem to show that cap sets must have size (see e.g. this paper of Meshulam). This of course is better than the trivial bound of once n is large. In the converse direction, the trivial example shows that cap sets can be as large as ; the current world record is , held by Edel. The gap between these two bounds is rather enormous; I would be very interested in either an improvement of the upper bound to , or an improvement of the lower bound to . (I believe both improvements are true, though a good friend of mine disagrees about the improvement to the lower bound.)
One reason why I find this question important is that it serves as an excellent model for the analogous question of finding large sets without progressions of length three in the interval . Here, the best upper bound of is due to Bourgain (he also has a recent, not yet published, improvement to , while the best lower bound of is an ancient result of Behrend. Using the finite field heuristic that “behaves like” , we see that the Bourgain bound should be improvable to , whereas the Edel bound should be improvable to something like . However, neither argument extends easily to the other setting. Note that a (still open) conjecture of Erdős-Turán is essentially equivalent (for progressions of length three, up to log log factors) to the problem of improving the Bourgain bound to .
The Roth bound of appears to be the natural limit of the purely Fourier-analytic approach of Roth, and so any breakthrough would be extremely interesting, as it almost certainly would need a radically new idea. The lower bound might be improvable by some sort of algebraic geometry construction, though it is not clear at all how to achieve this.
(Update, Feb 25: After some feedback and advice, and moving the entire blog to another site, I have finally gotten the math formulae to work out nicely. Thanks for all the help!)
(Update, Feb 27: As pointed out in the comments, one can interpret this problem in terms of the wonderful game Set, in which case the problem is to find the largest number of cards one can put on the table for which nobody has a valid move. As far as I know, the best bounds on the cap set problem in small dimensions are the ones cited in the Edel paper mentioned above.)
(Update, Mar 5: After discussions with Jordan Ellenberg, we realised that there is a variant formulation of the problem which may be a little bit more tractable. Given any , the fewest number of lines in a set of of density at least is known to be for some ; this is essentially a result of Croot. The reformulated question is then to get as strong a bound on ) as one can. For instance, the counterexample shows that , while the Roth-Meshulam argument gives .)