I first learned about this problem from Éric Balandraud. So I wrote to him a couple of days ago, and he has just sent me an e-mail explaining that the question dates back (at least) to 1971. Éric attributes it to Erdős and Graham: He hasn't provided me with a reference (he is in the desert these days), but I will come back and update this answer as soon as I hear from him again.

I first learned about this problem from Éric Balandraud (in 2013). So I wrote to him a couple of days ago, and he has just sent me an e-mail explaining that the question dates back (at least) to 1971. Éric attributes it to Erdős and Graham: He hasn't provided me with a reference (he is in the desert these days), but I will come back and update this answer as soon as I hear from him again.

Update (March 31, 2017). I've just got a new message from Éric Balandraud. Below is an excerpt.

Here is the reference:

R. Graham, On sums of integers taken from a fixed sequences, Proc. Wash. State Univ. Conf. on Number Theory (1971), 22-40.

The question is stated as Conjecture 10 on page 36. It also appears in:

P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory, Monographie N28 de L'Enseignement Mathématiques, Université de Genève (1980).

The conjecture is stated there on page 95.