Improved bounds for Erdősʼ Matching Conjecture

The main result is the following. Let F be a family of k-subsets of an n-set, containing no s + 1 pairwise disjoint edges. Then for n ⩾ ( 2 s + 1 ) k − s one has | F | ⩽ ( n k ) − ( n − s k ) . This upper bound is the best possible and confirms a conjecture of Erdős dating back to 1965. The proof is surprisingly compact. It applies a generalization of Katonaʼs Intersection Shadow Theorem.