A note on the Erd{o}s Matching Conjecture

The Erd os Matching Conjecture states that the maximum size $f(n,k,s)$ of a family F⊆ [n]k that does not contain s pairwise disjoint sets is max\|Ak,s|,|Bn,k,s|\, where Ak,s=[sk-1]k and Bn,k,s=∈ [n]k:B∩ [s-1]≠ ∅\. The case $s=2$ is simply the Erd{o}s-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+cs,k] 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+1\. For $n=sk$, we show the stability of the unique extremal construction of size sk-1k=s-1/sskk with respect to minimal degree. As a consequence we derive limk→ ∞f(sk+1,k,s)sk+1k<s-1/s-εₛ for some positive constant εₛ which depends only on s.