A Similarity Space Approach to the 1/3-2/3 Conjecture in Partially Ordered Sets

Abstract

The 1/3-2/3 Conjecture, a longstanding open problem in combinatorics, posits that in any non-chain finite partial order, there exists a pair of elements (x, y) such that the probability P(x ≺ y) lies in the interval [1/3, 2/3]. This paper presents a novel approach to addressing this conjecture by leveraging the concept of similarity spaces. We introduce the notion of a ’balance constant’ within the framework of similarity spaces and prove that its supremum is 1/2, confirming a related conjecture. Our main contribution is the proof that the infimum of normalized similarity in posets is 1/3, achieved in the case of total disorder. This result provides a new perspective on the 1/3-2/3 Conjecture, connecting it to the theory of similarity spaces. We also establish relationships between edit distance, swap operations, and Generalized Rozinek Similarity, offering insights into the structure of totally disordered sequences. While we provide a proof of the 1/3-2/3 Conjecture in the general case using our approach, we acknowledge that a more rigorous and detailed proof is still needed to fully resolve this longstanding problem in combinatorics and order theory.