Beck–Fiala theorem

Statement

In mathematics, the Beck–Fiala theorem is a major theorem in discrepancy theory due to József Beck and Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately the same number of elements of each color. The Beck–Fiala theorem is concerned with the case where each element doesn't appear many times across all sets. The theorem guarantees that if each element appears at most $t$ times, then the elements can be colored so that the imbalance is at most $2t - 1$. Beck and Fiala conjectured that the imbalance can be even bounded by $O(\sqrt t)$.