Evaluation of the Komlos Conjecture Using Multi-Objective Optimization

Abstract

The Komlos conjecture, which explores the existence of a constant upper bound in the realm of n-dimensional vectors, specifically addresses the function K(n). This function, intricately defined as

encapsulates the maximal discrepancy within a set of n -dimensional vectors. This paper endeavors to unravel the mysteries of K(n), by meticulously evaluating its behavior for lower dimensions . Our findings revealed through systematic exploration, showcase intriguing values such as , , ,  and , shedding light on the intricate relationships within n-dimensional spaces. Venturing into higher dimensions, we introduce the function as a potentially robust lower bound for K(n). This innovative approach aims to provide a deeper understanding of the limiting behavior of K(n) as the dimensionality expands. As a culmination of our comprehensive analysis, we arrive at a significant revelation the Komlos conjecture stands refuted. This conclusion stems from the suspected divergence of K(n), as n approaches infinity, as evidenced by . This seminal result challenges established notions and added a valuable dimension to the ongoing discourse in optimization and discrepancy theory.