Komlós discrepancy conjecture

Prove that every n by n matrix whose columns have Euclidean norm at most one admits a sign vector x in {-1,1}^n such that the infinity norm of Ax is bounded by an absolute constant.

Background

The Komlós conjecture is presented as a conjectural extension of Spencer’s theorem. Spencer’s theorem assumes that all matrix entries are bounded in absolute value, whereas the Komlós formulation imposes only an ℓ2-norm bound on each column and seeks a dimension-independent discrepancy bound.

References

Before concluding our discussion of discrepancy theory, it is impossible not to mention the beautiful and conjectural extension of Spencer's theorem known as the Koml\'{o}s conjecture, which says that one only needs to control the $\ell_2$ norm of the columns of the matrix $A$ to arrive at the same conclusion as Spencer's theorem (the normalization is changed here to match the literature).

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