Erdős Problem #178

Let \(A_1,A_2,\ldots\) 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 \[\max_{m, 1\leq i\leq d} \left\lvert \sum_{1\leq j\leq m} f(a_{ij})\right\rvert \ll_d 1\] for all \(d\geq 1\)? Erdős remarks 'it seems certain that the answer is affirmative'. This was solved by Beck [Be81]. Recently Beck [Be17] proved that one can replace \(\ll_d 1\) with \(\ll d^{4+\epsilon}\) for any \(\epsilon>0\).