The expectation threshold 
of an increasing family on a
finite set 
is the supremum of the densities
at which the family has a containment cover of cost at
most 
.
Writing 
,
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 expectation value
of the number of members of 
contained in 
is the sum above.
Allowing nonnegative fractional weights on the witness sets gives the fractional
expectation threshold 
. The conjecture of Talagrand
is that 
for a universal constant 
(Dubroff et al. 2026). Dubroff et al. (2026)
prove the restricted statement in which the fractional witness is supported on sets of bounded size, with a constant depending on that bound.
