Vector balancing is the problem of assigning signs to a given collection of vectors so that their signed sum has a small prescribed vector norm. A permutation variant also allows the coordinates of each vector to be independently permuted (Niles-Weed et al. 2026).
The Komlós conjecture asks for a bound independent of the number of vectors and the dimension
when their Euclidean norms are at most 1 and the
signed sum is measured in the infinity norm. Guo
et al. (2026) report a bound less than 
, and Karingula and Lovett (2026) give an elementary
proof with bound 36. These statements concern the final
signed sum with signs alone. As of Oct. 26, 2026, journal peer review of the
cited 2026 proofs had not been verified.
Niles-Weed et al. (2026) proved that, for any vectors 
, 
, ..., 
with Euclidean norm
at most 1, there are signs 
and coordinate permutations 
such that
 |
where 
is the infinity norm. The choices can be made online,
using only the vectors seen so far. The bound tends to
1 as 
and is asymptotically optimal, since a single coordinate unit
vector forces a bound of at least 1. The result concerns the variant with permutations, not sign choices alone.
