Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA1, ...,Ak are distinct subsets of a set of cardinalityn, and |Ai ∩Aj|≦1 for 1≦in, then some two ofA1, ...,Ak have empty intersection. We prove a strengthening, that at leastk /n ofA1, ...,Ak are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary.