**Komlos Conjecture**: the exists an absolute constant $K>0$ such that for all $d$ and any collection of vectors $v\_1,\ldots, v\_n\in \mathbb{R}^d$ with $\left\lVert v\_i\right\rVert \_2=1$ we can find weights $w\_i\in\{-1,1\}$ such that $\left\lVert w\_1v\_1+\cdots+w\_nv\_n\right\rVert \_{\infty}<K$.


I would like to ask what the current state of this conjecture is? Is it true that the best result towards this conjecture is that for fixed $d$ one can show the latter statement with $K=K(n)\approx \log n$?



Current state of the Komlos conjecture on vector balancing
For fixed $d$, one can actually achieve a bound independent of $n$. More precisely, $K=K(d)=O(\sqrt{d})$ is fine, uniformly in $n$.


Proof : the unit ball of $\mathbb{R}^d$ can be covered by $2C^d$ balls of radius $\frac{1}{2}$, for some absolute constant $C$. Let $v\_1,\dots,v\_n$ be $n$ vectors in $\mathbb{R}^d$ of norm at most $1$. If $n>2C^d$, then one can find $v\_i,v\_j$, for some $i<j$, belonging to the same ball of radius $\frac{1}{2}$; in particular $||v\_i-v\_j||\_2 \leq 1$. By considering, the finite list $v\_i- v\_j, v\_1,\dots,\hat{v\_{i}},\dots,\hat{v\_j}, \dots v\_n$, one reduces to the case of $n-1$ vectors of norm at most $1$. Iterating this process, one reduces to the case $n \leq 2C^d$. Then one applies Banaszczyk's result to get a bound $O(\log^{\frac{1}{2}} 2C^{d}) = O(\sqrt{d})$.


