What are the largest known lower bounds for $B_k$, the maximal sum of the reciprocals of the members of subsets of the positive integers which contain no arithmetic progressions of length $k$? for $k=3,4,5,6...$
$B_k\leq$ sup({ $\sum_{n\in S}1/n$ |$S\subset N$|S contains no arithmetic progressions of length k})
And is the bound proved to be finite for any k?
Can there exist a subset for which the maximal bound (finite or infinite), is actually attainable?
Ok, I am interested in any known bounds on $B_k$