Article
Version 1
Preserved in Portico This version is not peer-reviewed
The Disprove of the Komlos Conjecture
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)
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)
How to cite: Brahim Belhaouari, S.; AlQudah, R. The Disprove of the Komlos Conjecture. Preprints 2019, 2019020059. https://doi.org/10.20944/preprints201902.0059.v1 Brahim Belhaouari, S.; AlQudah, R. The Disprove of the Komlos Conjecture. Preprints 2019, 2019020059. https://doi.org/10.20944/preprints201902.0059.v1
Abstract
Komlos conjecture is about the existing of a universal constant K such that for all dimension n and any collection of vectors (V_1 ) ⃗ ,…,(V_n ) ⃗ ∈R^n with ‖(V_i ) ⃗ ‖_2≤1 , there are weights ε_i∈{-1,1} in such that ‖∑_(i=1)^n▒〖ϵ_i (V_i ) ⃗ 〗‖_∞≤K(n)≤K. In this paper, the constant K(n) is evaluated for n≤5 to be K(2)=√2, K(3)=(√2+√11)/3, K(4)=√3, and K(5)=(4+√142)/9. For higher dimension, the function f(n)=√(n-⌈〖 log〗_2 (2^(n-1)⁄n)⌉ ) is found to be the lower bound for the constant K(n), from where it can be concluded that the Komlos conjecture is false i.e., the universal constant K=max┬(‖(V_i ) ⃗^* ‖_2≤1)min┬(ε_i=±1)〖‖∑_(i=1)^n▒〖ϵ_i (V_i ) ⃗^* 〗‖_∞ 〗 does not exsit because of lim┬(n→∞)〖K(n)≥lim┬(n→∞)√(log(n)-1)=+∞〗.
Keywords
Komlos Conjecture; optimazation; discrepancy theory.
Subject
Computer Science and Mathematics, Computational Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment
