A cover-free family of parameter 
, or 
-cover-free family, is a family 
of distinct nonempty
sets such that no member is contained in the union
of at most 
other members. This condition limits how efficiently the members can cover one another
(Huang and Liu 2026).
Let 
be the least size 
of a ground set admitting an 
-cover-free family with more than 
members. Huang and Liu (2026) proved
 |
for every integer 
. Together with the known upper bound from affine
planes, this gives 
as 
, resolving the asymptotic Erdős-Frankl-Füredi
conjecture (Huang and Liu 2026). This asymptotic result does not give an exact formula
for 
.
Huang and Liu (2026) state that the mathematical ideas and proofs
are their own, while an AI tool suggested a simplification of calculations.
Hu and Zhang (2026) strengthened the lower bound to
 |
for every integer 
, proving the strong Erdős-Frankl-Füredi cover-free
conjecture. The affine plane construction attains
this value whenever 
is a prime power, so 
in those cases. Hu and Zhang (2026) report AI assistance
during early proof development.
