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Komlós conjecture (constant ℓ∞ discrepancy for unit-norm columns)

Establish the Komlós conjecture: Determine a universal constant C such that for every m×n real matrix M whose columns have ℓ2-norm at most 1, there exist signs x in {−1,1}^n with ||Mx||_∞ ≤ C, independent of m and n.

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Background

The Komlós conjecture is a central question in discrepancy theory and generalizes the Beck–Fiala setting. It asks for a dimension- and size-independent constant bound on the ℓ∞-discrepancy when columns have unit ℓ2-norm.

Non-constructively, Banaszczyk established an O(√log n) bound; more recently, Bansal and Jiang improved the bound to O(log{1/4} n). This paper contributes a related local mean squared discrepancy guarantee for row subsets, but the conjectured absolute-constant ℓ∞ bound remains open.

References

The central open problems in this area are the Beck-Fiala conjecture and more generally the Komlos conjecture, where given a set of $n$ vectors in the unit ball in $m$, the goal is to assign signs (also called ``colors") to the vectors so as to bound the $\ell_{\infty}$ norm of the final signed sum, by a constant independent of $m$ and $n$.

— Constructive l2-Discrepancy Minimization with Additive Deviations (2508.21423 - Dutta, 29 Aug 2025) in Section 1, Introduction, paragraph “Discrepancy Minimisation and The Bansal-Garg Algorithm”