Let $A_1,A_2,\dots$ be an infinite collection of infinite sets of integers, say $A_i=\{a_{i1}<a_{i2}<\cdots \}$. Does there exist some $f:\mathbb{N}\to \{-1,1\}$ such that$$\underset{m,1\le i\le d}{\max }\left|\sum _{1\le j\le m}f(a_{ij})\right|{\ll }_{d}1$$for all $d\ge 1$?