preprints.org > computer science and mathematics > computational mathematics > doi: 10.20944/preprints201902.0059.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 5 February 2019 / Approved: 6 February 2019 / Online: 6 February 2019 (10:12:50 CET)
Version 2 : Received: 26 June 2019 / Approved: 9 July 2019 / Online: 9 July 2019 (08:41:45 CEST)
Version 3 : Received: 30 October 2019 / Approved: 31 October 2019 / Online: 31 October 2019 (06:09:49 CET)

Komlos conjecture is about the existing of a universal constant $K$ such that for all dimension $n$ and any collection of vectors $\overrightarrow{V_1},\dots ,\overrightarrow{V_n}\in {ℝ}^n$ with $\overrightarrow{V_{.}}{}_2\le 1$ , there are weights ${\epsilon }_i\in \left\{-1,1\right\}$ in such that${\displaystyle \sum }_{i=1}^n$. In this paper, the constant $K\left(n\right)$ is evaluated for $n\le 5$ to be $K\left(2\right)=\sqrt{2}$, $K\left(3\right)=\frac{\sqrt{2}+\sqrt{11}}{3}$, $K\left(4\right)=\sqrt{3}$, and $K\left(5\right)=\frac{4+\sqrt{142}}{9}$. For higher dimension, the function $f\left(n\right)=\sqrt{n-lo{g}_2\left(2^{n-1}/n\right)}$ is found to be the lower bound for the constant $K\left(n\right)$, from where it can be concluded that the Komlos conjecture is false i.e., the universal constant $K$ does not exist because of $\underset{\text{n}\to \infty }{\lim }K\left(n\right)\ge \underset{\text{n}\to \infty }{\lim }\sqrt{\log \left(n\right)-1}=+\infty$.

Komlos Conjecture; optimazation; discrepancy theory

Computer Science and Mathematics, Computational Mathematics

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