Komlós conjecture: necessity of the √log n factor

Determine whether there exists a universal constant C such that for every n and every matrix A ∈ ℝ^{n×n} with each row having ℓ2 norm at most 1, there exists ε ∈ {−1,1}^n with ||Aε||∞ ≤ C; equivalently, ascertain whether the √(log n) factor in Banaszczyk’s vector balancing bound can be removed and replaced by an absolute constant.

Background

The paper discusses the relationship between the bad science matrix problem and classic vector balancing questions, highlighting Banaszczyk’s bound that guarantees a sign vector with ℓ∞ norm O(√log n). The authors explicitly point out the central open question: whether the √log n factor is necessary, which is the Komlós conjecture.

This problem is external to the bad science matrix problem but is included as a motivating backdrop; resolving it would clarify whether the logarithmic factor is inherent to all such vector balancing settings.

References

It is known (see Banaszczyk ) that there exists a constant $C>0$ such that for all matrices $A \in \mathbb{R}{n \times n}$ whose rows are of size $\leq 1$ in $\ell2$, there always exists a sign vector $\varepsilon \in \left{-1, 1 \right}n$ such that $$ |A \varepsilon|_{\ell{\infty} \leq C \sqrt{\log{n}.$$ The open question is whether the $\sqrt{\log{n}$ term is necessary or whether the result remains true with a universal constant $C$.

— On the Structure of Bad Science Matrices (2408.00933 - Albors et al., 1 Aug 2024) in Subsection “The Komlós conjecture,” Introduction