Beck–Fiala conjecture for regular hypergraphs

Let $\mathcal{H}$ be a $t$-regular hypergraph, meaning that every vertex belongs to exactly $t$ hyperedges. For a hypergraph $\mathcal{H}=(V,E)$, let $\operatorname{disc}(\mathcal{H})$ denote its discrepancy, the minimum over all $\pm1$ colorings of the maximum absolute signed edge sum.

Beck–Fiala conjecture. For a $t$-regular hypergraph $\mathcal{H}$, we have

$$\operatorname{disc}(\mathcal{H})=O(\sqrt{t}).$$

This is one of the central open problems in discrepancy theory; the paper’s main spectral bound gives an $O(\sqrt{t}+\lambda)$ estimate, but does not establish the conjectured bound for general $t$-regular hypergraphs.