Families with no matchings of size s

Abstract Let k ⩾ 2 , s ⩾ 2 be positive integers. Let [n] be an n-element set, n ⩾ k s . Subsets of 2 [ n ] are called families. If F ⊂ ( [ n ] k ) , then it is called k-uniform. What is the maximum size e k ( n , s ) of a k-uniform family without s pairwise disjoint members? The well-known Erdős Matching Conjecture would provide the answer for all n, k, s in the above range. For n > 2 k s it is known that the maximum is attained by A 1 ( T ) : = { A ⊂ [ n ] : | A | = k , A ∩ T ≠ ∅ } for some fixed ( s − 1 )-element set T ⊂ X . We discuss recent progress on this problem. In particular, our recent stability result states that for n > ( 2 + o ( 1 ) ) k s and a k-uniform family F , F ⊈ A 1 ( T ) , then | F | is considerably smaller. This result is applied to obtain the corresponding anti-Ramsey numbers in a wide range. Removing the condition of uniformness, we arrive at another classical problem of Erdős, which was solved by Kleitman for n ≡ 0 or −1 (mod s). We succeeded in resolving this long-standing problem for n ≡ − 2 ( mod s ) via a new averaging technique which might prove useful in various other situations.