The Shareshian-Wachs conjecture asserts elementary positivity, together with unimodality, for chromatic quasisymmetric functions
of natural unit interval graphs. Such
a graph on ![[n]={1,...,n}](/images/equations/Shareshian-WachsConjecture/Inline1.svg)
is specified by weakly increasing integers 
, ..., 
with 
,
using edges 
for 
.
Its elementary-basis expansion is
 |
where 
ranges over the partitions of 
and 
,
with 
an elementary symmetric polynomial.
Elementary positivity requires each polynomial 
to have nonnegative integer coefficients.
Unimodality further requires each coefficient sequence
to weakly increase and then weakly decrease, making it a unimodal
sequence.
OpenAI (2026) announced coefficientwise elementary positivity for every natural unit interval graph. The reported positivity holds coefficientwise in 
, which is stronger than positivity after setting 
. Liu (2026) subsequently reported the stronger decomposition
 |
where 
is the number of edges, 
denotes the floor function,
and the 
are nonnegative integers. This gives elementary
positivity and unimodality for every natural
unit interval graph and every partition 
of its number of vertices.
Liu's argument uses the framework in OpenAI's manuscript rather than deriving unimodality
from positivity alone. OpenAI reports an AI-generated proof.
Liu reports GPT-6 Pro and Claude Fable assistance with mathematical exploration,
proof development, and checking. As of Nov. 3, 2026,
independent specialist verification of the complete claimed proofs
had not been reported.
