Matching of Given Sizes in Hypergraphs

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer (which may not divisible by $k$), $s\le \lfloor n/k\rfloor-1$. We show if $H$ is $k$-uniform hypergraph on vertices with $\delta_{d}(H)>\binom{n-d}{k-d}-\binom{n-d-s+1}{k-d}$, then contains matching of size $s$. This improves recent result Lu, Yu, Yuan also answers question Kühn, Osthus, Townsend. In many cases, our can strengthened to $s\leq n/k\rfloor$, which covers the entire possible range On other hand, there are examples showing does hold for certain $n, k, d$, $s= n/k\rfloor$.