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 be divisible by k), and let s \leq \lfloor n/k\rfloor - 1. We show that if H is a k-uniform hypergraph on n vertices with \delta _d(H) > ^\bigl(n_{k -}^-_d^d\bigr) - ^{\bigl(n -}_k^d_-^-_d^s+1\bigr) , then H contains a matching of size s. This improves a recent result of Lu, Yu, and Yuan and also answers a question of K\" uhn, Osthus, and Townsend. In many cases, our result can be strengthened to s \leq \lfloor n/k\rfloor , which then covers the entire possible range of s. On the other hand, there are examples showing that the result does not hold for certain n, k, d, and s = \lfloor n/k\rfloor .