Breaking the Infinite Barrier in the $\frac{1}{3}$--$\frac{2}{3}$ Conjecture

2026-09-25

AI Summary

This study addresses the $\frac{1}{3}$-$\frac{2}{3}$ conjecture on the balance constant of partially ordered sets (posets), which has seen no substantial progress since Brightwell et al. established general bounds in 1995. By integrating tools from combinatorics and probability theory, this work provides an in-depth analysis of the distributional properties governing the relative order of elements in linear extensions of posets, thereby transcending the limitations of existing theoretical frameworks. The primary contribution is the first general improvement to this conjecture in nearly three decades, elevating the lower bound of the balance constant by a small quantity $\varepsilon > 0$. This breakthrough resolves a long-standing stagnation in the field and offers critical new progress toward the ultimate resolution of the $\frac{1}{3}$-$\frac{2}{3}$ conjecture.

Abstract

The balance constant is a poset parameter measuring how well random linear extensions can be split according to their relative order on two elements. We give an $\varepsilon$-improvement for the balance constant over the bound by Brightwell, Felsner and Trotter (1995), for some small $\varepsilon>0$. This is the first general result towards the $\frac{1}{3}$--$\frac{2}{3}$ conjecture in over 30 years.

Problem

• balance constant
• poset
• 1/3-2/3 conjecture
• linear extensions

Innovation

• balance constant
• 1/3-2/3 conjecture
• linear extensions
• poset
• epsilon-improvement