Let 
be a finite set and let 
be an increasing family.
The family is 
-small if there is a collection 
such that every member of
contains a member of 
and
 |
The expectation threshold is
 |
The name comes from the first-moment obstruction: in the binomial random subset 
, the expected number of members of 
contained in 
is the sum above.
Allowing nonnegative fractional weights on the witness sets gives the fractional
expectation threshold 
. Talagrand conjectured that 
for a universal constant 
. Dubroff, Kahn, and Park (2026) prove
the restricted statement in which the fractional witness is supported on sets
of bounded size, with a constant depending on that bound.
