The Disprove of the Komlos Conjecture

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 . → 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− lo g 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 does not exist because of lim n→∞ K( n )≥ lim n→∞ log( n )−1 =+∞.