Packing nearly-disjoint sets

De Bruijn and Erdo{combining double acute accent}s proved that if A ₁, ..., A _k are distinct subsets of a set of cardinality n, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, and k>n, then some two of A ₁, ..., A _k have empty intersection. We prove a strengthening, that at least k /n of A ₁, ..., A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdo{combining double acute accent}ds, Faber and Lovász of which it is a corollary.