Everyone knows about the problem of finding the largest set in 1,2,...,N which contains no arithmetic progression of length 3.....what happens if we look at the set which is the smallest maximal subset of 1,...,N which contains no 3-AP?

Obviously we have a square root lower bound using a greedy argument and a simple upper bound, with exponent log(2)/log(3), by looking at integers with no 2 in their base 3 representation. Is anything better known, in either direction? I know about the original work of Stanley and Odlyzko.......