Beck–Fiala theorem and conjecture

**The Beck–Fiala theorem**

If $S_1, \dots, S_m \subseteq [n]$ is a set system of degree at most $t$, i.e. every $j \in [n]$ lies in at most $t$ of the sets, and $t \ge 1$, then there is a colouring $\chi \colon [n] \to \{-1, +1\}$ with $\left|\sum_{j \in S_i} \chi(j)\right| \le 2t - 1$ for every $i$.

The hypothesis $t \ge 1$ is necessary: a system of degree $0$ consists of empty sets only, whose discrepancy is $0 > 2 \cdot 0 - 1$.

[J. Beck and T. Fiala, *"Integer-making" theorems*, Discrete Applied Mathematics **3** (1981), 1–8.]