A note on the Erdos Matching Conjecture

The Erd os Matching Conjecture states that the maximum size f (n, k, s) of a family F nk that does not contain s pairwise disjoint sets is \|A₊, ₒ|, |B₍, ₊, ₒ|\, where A₊, ₒ=sk-1k and B₍, ₊, ₒ=\B n{k: B s-1 \}. The case s=2 is simply the Erdos-Ko-Rado theorem on intersecting families and is well understood. The case n=sk was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if k, s are fixed and n is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for n sk, sk+cₒ, ₊ if s is large enough compared to k. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most s in a similar range of parameters. In this short note, we are concerned with the case s 3 fixed, k tending to infinity and n\sk, sk+1\. For n=sk, we show the stability of the unique extremal construction of size sk-1k=s-1sskk with respect to minimal degree. As a consequence we derive ₊ f (sk+1, k, s) sk+1{k}<s-1s-ₛ for some positive constant ₛ which depends only on s.