[math-fun] Erdos conjecture on arithmetic progressions

Erdos conjectured that if the reciprocal sum of a sequence of positive integers diverges, then it contains k-arithmetic progressions for any k. It is known* that this implies the existence of constants N_k such that if the sum of the reciprocals of a sequence is greater than N_k, the sequence contains a k-arithmetic progression. Of all subsets of positive integers not containing a 3-arithmetic progression, what's the highest reciprocal sum you can find? The greedy method gets over 3.0078. * Gerver 1977, Brown & Freedman 1987 in a stronger form Charles Greathouse Analyst/Programmer Case Western Reserve University