arXiv:2609.30888 · v2

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

首次提交 2026-09-25 14:53

版本更新 2026-10-29 03:02

雷达观测 2026-10-30 10:22

摘要

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.

作者备注：32 pages, v2 simplifies the proof of Lemmas 4.2 and 6.4