Erdős Problem #178 — As far as we know the following related more general problem is still open.

As far as we know the following related more general problem is still open. Let $A_k = \{a_1^{(k)} < a_2^{(k)} < ...\}$, $k = 1, 2, ...$ be an infinite class of infinite sets of integers. Does there exist a function $F(d)$ (depending on the sequences $A_k$) so that for a suitable $g(n) = \pm 1$

$$\max_{m, 1 \leq k \leq d}\left|\sum_{i=1}^{m} g(a_i^{(k)})\right| < F(d)\ ?$$

It seems certain that the answer is affirmative.