Preprint Article Version 1 This version is not peer-reviewed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)

How to cite: Brahim Belhaouari, S.; AlQudah, R. The Disprove of the Komlos Conjecture. Preprints 2019, 2019020059 (doi: 10.20944/preprints201902.0059.v1). Brahim Belhaouari, S.; AlQudah, R. The Disprove of the Komlos Conjecture. Preprints 2019, 2019020059 (doi: 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)=+∞〗.

Subject Areas

Komlos Conjecture; optimazation; discrepancy theory.

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