Erdos offered a 
prize for a proof of the proposition that "If the sum of reciprocals of a set of integers diverges, then that set contains arbitrarily long arithmetic progressions." This conjecture is still open (unsolved), even for 3-term arithmetic progressions. Erdos also offered 
for an asymptotic formula for 
,
that does not contain a 3-term arithmetic progression.
A-Sequence, B2-Sequence, Szemerédi's Theorem
Erdos, P. and Turán, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261-264, 1936.
Green, B. and Tao, T. "The Primes Contain Arbitrarily Long Arithmetic Progressions." Preprint. 8 Apr 2004. http://arxiv.org/abs/math.NT/0404188.