A sharp version of Talagrand's selector process conjecture, with applications to rounding fractional covers and Bernoulli Sudakov minoration

We prove a sharp version of Talagrand's selector process conjecture. Roughly speaking, given any collection of nonnegative weight vectors whose support form a family that is not $p$-small, a random set of density $O(sp)$ captures all but a $2^{-s}$ fraction of weight of some vector with high probabi