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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 $B_k$

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    Please edit the question to add that reference.2012-02-28

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Gerver, Joseph L., The sum of the reciprocals of a set of integers with no arithmetic progression of k terms, Proc. Amer. Math. Soc. 62 (1977), no. 2, 211–214, MR0439796 (55 #12678) proves for every $\epsilon\gt0$, there exist, for all but a finite number of integers $k\ge3$, $k$-free sets $S_k$ for which $\sum_{n\in S_k}1/n\gt(1−\epsilon)k\log k$. (A $k$-free set is one with no $k$-term arithmetic progression.)

Wróblewski, J., A nonaveraging set of integers with a large sum of reciprocals, Math. Comp. 43 (1984), no. 167, 261–262, MR0744935 (85k:11006) constructs a set with no three elements in arithmetic progression such that the sum of the reciprocals of the elements is greater than 3.00849.

Pal, Goutam, Sequences of positive integers containing no $p$ terms in arithmetic progression generated by the greedy algorithm, J. Ramanujan Math. Soc. 24 (2009), no. 3, 249–263, MR2568055 (2011b:11020) might also be of interest.