Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5777111 | Electronic Notes in Discrete Mathematics | 2017 | 7 Pages |
Let k⩾2, s⩾2 be positive integers. Let [n] be an n-element set, n⩾ks. Subsets of 2[n] are called families. If Fâ([n]k), then it is called k-uniform. What is the maximum size ek(n,s) of a k-uniform family without s pairwise disjoint members? The well-known ErdÅs Matching Conjecture would provide the answer for all n, k, s in the above range. For n>2ks it is known that the maximum is attained by A1(T):={Aâ[n]:|A|=k,Aâ©Tâ â } for some fixed (sâ1)-element set TâX. We discuss recent progress on this problem. In particular, our recent stability result states that for n>(2+o(1))ks and a k-uniform family F,FâA1(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 ErdÅs, which was solved by Kleitman for nâ¡0 or â1 (mod s). We succeeded in resolving this long-standing problem for nâ¡â2(mods) via a new averaging technique which might prove useful in various other situations.