An Exposition of the $\widetilde{O}(\log^{1/4} n)$ Bound for the Komlós Problem

2026-08-28

AI Summary

本文解决了Komlós问题，通过证明任何列向量欧几里得范数不超过1的矩阵的组合不均衡性上限为O((log n)^(1/4)(log log n)^(7/4))，改进了先前的O(√log n)界限。

Abstract

A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.

Problem

• combinatorial discrepancy
• matrix
• Euclidean norm

Innovation

• combinatorial discrepancy
• asymptotic improvement
• Euclidean norm