Preprint Article Version 2 This version is not peer-reviewed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)

How to cite: Brahim Belhaouari, S.; AlQudah, R. The Disprove of the Komlos Conjecture. Preprints 2019, 2019020059 (doi: 10.20944/preprints201902.0059.v2). Brahim Belhaouari, S.; AlQudah, R. The Disprove of the Komlos Conjecture. Preprints 2019, 2019020059 (doi: 10.20944/preprints201902.0059.v2).

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 n with V . 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 n5 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 lo g 2 ( 2 n1 /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 does not exist because of lim n K( n ) lim n log( n )1 =+.

Subject Areas

Komlos Conjecture; optimazation; discrepancy theory

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