The Komlós conjecture is the assertion that there is an absolute constant 
such that, for all positive
integers 
and 
and all vectors 
, ..., 
with Euclidean norm
at most 1, there are signs 
satisfying
 |
where 
is the infinity norm and 
is independent of both 
and 
(Karingula and Lovett 2026). This is a vector
balancing problem with signs alone. Banaszczyk (1998) proved a bound of order
in place of the absolute constant.
Guo et al. (2026) report a proof with the strict bound
 |
They attribute discovery of the proof to the Odin Automatic AI Research Agent. Karingula and Lovett (2026) give an elementary proof of the same conjecture with bound 36 by simplifying the argument of Guo et al. (2026). As of Oct. 26, 2026, journal peer review of these proofs had not been verified.
The assertion bounds the final signed sum. It does not require a simultaneous bound on all partial sums and does not permit coordinate permutations. The stated constants are upper bounds, not assertions of optimality.
-Dimensional
Convex Bodies." Random Struct. Alg. 12, 351-360, 1998.