Beck–Fiala discrepancy conjecture

Prove that every finite hypergraph in which each vertex has degree at most d admits a two-coloring whose discrepancy on every hyperedge is at most C√d for an absolute constant C.

Background

The Beck–Fiala conjecture is described as the hypergraph-coloring companion to the Komlós conjecture. The desired bound is proportional to the square root of the maximum vertex degree, whereas the classical Beck–Fiala result gives the weaker bound 2d-1.

References

There is also a famous hypergraph colouring ``companion'' to this conjecture, made independently by Beck and Fiala .

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 2, subsection “The Komlós Conjecture and the Beck-Fiala conjecture”

Their proof is existential; the sharp algorithmic problem remains open and is the focus of this article.

— Improved Algorithms for Beck--Fiala with Bounded Sets  (2609.19714 - Altschuler, 17 Sep 2026) in Remark in Section 1 (Introduction)