It is often claimed that Erdős discrepancy conjecture was formulated by him in 1932. In Tao's proof of the conjecture he says the original formulation is
Every sequence $f(1),f(2),\ldots$ taking values in $\{-1,+1\}$ has infinite discrepancy.
Tao defines discrepancy as $$\sup_{k,n\in\mathbb{N}}\lvert\sum_{j=1}^kf(jn)\rvert$$
I am looking for the original formulation if it exists and hopefully some of his motivations for it. I have found two publications by Erdős in 1932 but both are in Hungarian and they seem to be proofs rather than problem statements. Here and here.
Erdős writes in 1957 that he made the conjecture "Twenty-five years ago". So that's at least a reference by himself, but unfortunately not the desired original publication. As he doesn't seem to give a reference to such a publication there, I am beginning to suspect that the original conjecture might have been spread by "personal communication".