Disjointly representing set systems

A family F of sets is s-disjointly representable if there is a family S of disjoint sets each of size s such that every F∈F contains some S∈S. Let f(r,s) be the minimum size of a family F of r-sets which is not s-disjointly representable. We give upper and lower bounds on f(r,s) which are within a constant factor when s is fixed.