The classical discrepancy setting is as follows: Partition the vertices of a hypergraph into two ``colors'' (Red and Blue). The discrepancy of this partition is the maximum difference between the colors in any Hyperedge. The discrepancy of the hypergraph is the mimimum discrepancy for any partition of the vertices.

The Beck-Fiala conjecture states that for a $t$-regular hypergraph (every vertex is in $t$ hyperedges), the discrepancy is at most $O(\sqrt{t})$. The seeming difficulty in making any progress towards resolving this has led to the study of the dicrepancy of random $t$-regular hypergraphs with $n$ vertices and $m$ hyperedges. Here, Ezra and Lovett (RANDOM 2015) showed that the discrepancy is at most $O(\sqrt{t \log t})$ almost surely as $t$ grows and $n = O(m)$. More recently, a result of Bansal and Meka (SODA 2019) shows that the discrepancy is almost surely $O(\sqrt{t})$ almost surely as $n$ grows and $t = \Omega(\log^2 \log n)$.

We will see a proof that for every $t > 0$, the discrepancy of a random $t$-regular hypergraph is almost surely $O(\sqrt{t})$ as $n$ grows and $n = O(m)$. This proof works more generally because it is based on a spectral ``pseudorandomness'' condition of the incidence matrix of the hypergraph, which holds with high probability.

Reference: https://arxiv.org/abs/1907.04117