English

The 1/3-2/3 Conjecture for ordered sets whose cover graph is a forest

Combinatorics 2017-06-20 v3

Abstract

A balanced pair in an ordered set P=(V,)P=(V,\leq) is a pair (x,y)(x,y) of elements of VV such that the proportion of linear extensions of PP that put xx before yy is in the real interval [1/3,2/3][1/3, 2/3]. We define the notion of a good pair and claim any ordered set that has a good pair will satisfy the conjecture and furthermore every ordered set which is not totally ordered and has a forest as its cover graph has a good pair.

Keywords

Cite

@article{arxiv.1610.00809,
  title  = {The 1/3-2/3 Conjecture for ordered sets whose cover graph is a forest},
  author = {Imed Zaguia},
  journal= {arXiv preprint arXiv:1610.00809},
  year   = {2017}
}
R2 v1 2026-06-22T16:09:33.929Z