Morris's conjecture (Morris 2006) concerns Frankl-complete configurations for the union-closed
sets conjecture. A configuration is a finite family 
of sets, and
its support 
is the set of elements occurring
in those sets. The family is a
Frankl-complete configuration, or
FC configuration, if every finite union-closed set 
containing 
has an element of 
belonging to at least half the members of 
. Thus an FC configuration is a local certificate for the conjecture:
whenever it occurs within a larger union-closed set,
one of the elements already in its support must be frequent.
A configuration that is not FC need not be a counterexample
to the conjecture, since a witnessing union-closed
set may have a frequent element outside 
. For fixed positive integers 
and 
, let 
be the least 
such that every family of 
distinct 
-element subsets of an 
-element ground set is a Frankl-complete
configuration. Morris's conjecture states that, for every fixed 
,
 |
as 
tends to infinity, where 
denotes big-theta
notation with constants allowed to depend on 
.
Liu (2026) reported a proof that the conjectured asymptotic relation holds for every fixed 
.
The upper bound uses recursively constructed Frankl-complete
configurations and extremal estimates, while an extension of Morris's ordered-block
construction gives a matching lower bound. For four-sets,
the paper gives 
,
where 
and 
denote big-O notation and little-o
notation, respectively. More explicitly, for 
and 
, where 
denotes the floor function,
 |
The proof also claims that a sunflower with a two-element core and disjoint two-element petals is a Frankl-complete configuration
exactly when it has at least nine petals, and admits a weighted certificate with
a positive margin exactly when it has at least ten petals. Liu (2026) also reports
and a counterexample to a lexicographic extremality
conjecture of Pulaj and Wood (2025).
The general argument and the finite classification were co-developed by Liu, GPT-6 Astra, and Fable 5.1, with the models making most of the finite classification. Exact certificates and verification programs accompany the finite results. Independent specialist review of the general proof and independent runs of the finite verifiers had not been reported as of Sep. 14, 2026.
