An elementary proof of the Komlós conjecture

Abstract

We give an elementary proof of the Komlós conjecture by simplifying the recent proof of Guo, Fang, and Lu. We show that any vectors $v_1,\dots ,v_n\in {\mathbb{R}}^d$ with $\Vert v_i{\Vert }_2\le 1$ admit signs ${\epsilon }_i\in \{-1,1\}$ such that $\Vert {\sum }_{i=1}^n{\epsilon }_i{v}_i{\Vert }_{\infty }\le 36$. The proof uses only elementary combinatorial and probabilistic arguments and basic calculus.

Expanded discussion of related works and added subsequent developments