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 
.
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.
