On the union of intersecting families

A family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdos raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ≥ 2 and for any , if X is an n-element set, and , where each is an intersecting family of k-element subsets of X, then , with equality only if for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis , in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.