Tim Gowers described two additional proposed polymath projects. One about the first unknown cases of the polynomial Density Hales Jewett problem. Another about the Littelwood’s conjecture.
I will state one problem from each of these posts:
1) (Related to polynomial DHJ) Suppose you have a family of graphs on n labelled vertices, so that we do not two graphs in the family such that is a subgraph of and the edges of which are not in form a clique. (A complete graph on 2 or more vertices.) can we conclude that =? (In other words, can we conclude that contains only a diminishing fraction of all graphs?)
Define the “distance” between two points in the unit cube as the product of the absolute value of the differences in the three coordinates. (See Tim’s remark below.)
2) (Related to Littlewood) Is it possible to find n points in the unit cube so that the “distance” between any two of them is at least ?
A negative answer to Littlewood’s problem will imply a positive answer to problem 2 (with some constant). So the pessimistic saddle thought would be that the answer to Problem 2 is yes without any bearing on Littlewood’s problem.