A remark on Kashin's discrepancy argument and partial coloring in the Koml\'{o}s conjecture

TL;DR

This paper revisits Kashin's early partial coloring result, showing it provides a simple proof for constant discrepancy partial coloring related to the Komlós conjecture, despite a logarithmic factor in Spencer's bound.

Contribution

It offers a simplified proof of constant discrepancy partial coloring in the context of the Komlós conjecture using Kashin's argument.

Findings

Kashin's argument yields a simple proof for partial coloring.

The result achieves constant discrepancy up to a log log n factor.

It connects Kashin's early work to modern discrepancy problems.

Abstract

In this expository note, we discuss an early partial coloring result of B. Kashin [C. R. Acad. Bulgare Sci., 1985]. Although this result only implies Spencer's six standard deviations [Trans. Amer. Math. Soc., 1985] up to a $\log\log n$ factor, Kashin's argument gives a simple proof of the existence of a constant discrepancy partial coloring in the setup of Koml\'{o}s conjecture.