Set systems with union and intersection constraints

Let 2 ⩽ d ⩽ k be fixed and n be sufficiently large. Suppose that G is a collection of k-element subsets of an n-element set, and | G | > ( n − 1 k − 1 ) . Then G contains d sets with union of size at most 2k and empty intersection. This extends the Erdős–Ko–Rado theorem and verifies a conjecture of the first author for large n.