Komlós conjecture

The Komlós conjecture*

There exists a universal constant $K > 0$ such that for all $n, m \in \mathbb{N}$ and all vectors $v_1, \dots, v_n \in \mathbb{R}^m$ with $\|v_i\|_2 \le 1$ (encoded here as $\sum_j v_{ij}^2 \le 1$), there exist signs $\varepsilon_i \in \{-1, +1\}$ such that $\left\|\sum_i \varepsilon_i v_i\right\|_\infty \le K$, i.e. $\left|\sum_i \varepsilon_i v_{ij}\right| \le K$ for every coordinate $j$.