The Komlos Conjecture Holds for Vector Colorings

The Komlós conjecture in discrepancy theory states that for some constant K and for any m× n matrix A whose columns lie in the unit ball there exists a vector x ∈ {−1,+1} such that ‖Ax‖∞ ≤ K. This conjecture also implies the Beck-Fiala conjecture on the discrepancy of bounded degree hypergraphs. Here we prove a natural relaxation of the Komlós conjecture: if the columns of A are assigned unit vectors in R rather than ±1 then the Komlós conjecture holds with K = 1. Our result rules out the possibility of a counterexample to the conjecture based on the natural semidefinite relaxation of discrepancy. It also opens the way to proving tighter efficient (polynomial-time computable) upper bounds for the conjecture using semidefinite programming techniques.